MID-June, late afternoon in Dundee and it’s almost ready. The barriers are still up, the fit-out is going on inside, but the structure is complete. Seven years’ work, £80.11m spent, 11,600 square metres covered. It sits low on the edge of the water, looming over the Tay. This is the museum that Kengo Kuma built.

Right now, getting on for 5pm, the Japanese architect is standing on a balcony in the Discovery Centre mere feet away, looking across at the building that he is responsible for and talking to me about the Scottish landscape, the randomness of nature and how much he loves whisky and haggis.

He is a quietly spoken, elegantly dressed man who would have appeared to have designed the museum in his own image. Distinctive but not overblown, expansive yet restrained.

Up close to the V&A Dundee you can’t help but be impressed; the curving rise and fall of it, the frozen waves of precast concrete slats. But from a distance you could miss it. Come over either of the Tay bridges – road or rail – and you have to look close to spot it on the waterfront.

In short, as signature buildings go, it’s either grandly modest or modestly grand. Or, maybe like the sharp blue and white suit Kuma-san is wearing today, it’s both subtle and showy at the same time.

What, I say, to the 63-year-old man standing beside me looking out at the museum and the Tay estuary, are we looking at?

The architect pauses to find the word in English, humming in hesitation before committing himself. “In the 20th century, the museum should be a white cube and should be very abstract and scientific, a space for showing art; a kind of laboratory of art,” he explains.” Now people want to go back to the natural environment. Even in the city people want to feel nature.

“I tried to learn many things from the forest and many things from the valley. The Scottish hill and valley is the most charming point of Scotland. I tried to bring that character to the building.

“In that sense,” he continues, “the building is intermediate between landscape and artefact. In 20th century architecture and nature were totally opposite. They were fighting. Now, we should become friends again. That is the basis of the building.”

In those few words Kuma has summed up the building and summed up his own approach to architecture. He is a starchitect (or, if he’s not you suspect he soon will be in the wake of the V&A Dundee and his upcoming stadium in Tokyo for the 2020 Olympics) who is in opposition to the whole idea of starchitecture, a man who speaks up for humility and sensuality, for texture and approachability, for natural materials and a sense of place. (The museum's interior is decked out in oak veneer panels.)

He is also the man who has designed the most important building constructed in Dundee for decades (at the very least), a building the city hopes will transform its economic, physical and possibly even psychological outlook. Kuma hopes so, too.

He talks about the V&A building as a “living room for the city” and, also, a gateway between the River Tay and the urban landscape that for so long had turned its back on it.

“That was my idea from the beginning of the project. We thought we should create a gate. We should not create a box. A box would separate nature and building.”

What he wanted to do was construct a building that could serve the same purpose as a Japanese Torii gate, traditionally a threshold between the sacred and the mundane.

His original idea had been to have one-third of the building in the Tay itself. But when tenders came in it was clear that the original £45m budget was nowhere near enough. And even though it eventually cost almost double that original budget there were still compromises required, with the building having to be pushed back into the land.

The question is how happy he is with the result? “Umm, in the process there was some stress,” he admits after consideration. “But, finally, I am satisfied 100 per cent.”

When he was 10 years old Kengo Kuma was taken by his father to see Tokyo’s new Olympic stadium. The Yoyogi National Gymnasium, shaped like a discus with a mast attached, was designed by Kenzo Tange for the 1964 games. Kuma was blown away by the beauty of the building.

“It was very different from other buildings in that period,” he recalls 54 years later. “In 1964 Tokyo was changing, but, still, most of the buildings were one-storey, two-storey, and most of the houses were made of wood. The Kenzo Tange building was outstanding. I was so shocked and so impressed I began to think: ‘Ah, the dream profession is architecture.’ That is the beginning of my story.”

But only the beginning. Because in many ways Kuma’s career has been a rejection of the Tange blueprint. Tange’s building, he says, was “very heroic, very strong, reaching the heavens.”

But in the years since he first saw the Tange stadium Kuma has become suspicious of the heroic gesture in architecture, has even written a book decrying it. He has spoken out for natural materials, for wood and stone. For some time, he has advocated architectural humility.

It took him a while to reach that point, it should be said. At university the “heroic gesture” was the thing to do.

“The education I got at university was not about wooden buildings,” he says. “That’s because the goal of the country was to achieve big buildings of concrete and steel to catch up America. That was the aim of the country. ‘The wooden building? Please forget. That is the past.’"

It became even more so when the starchitects arrived. In the 1980s and 1990s the likes of Norman Foster and Richard Rogers, Peter Zumthor and Zaha Hadid became brand names. Architecture was hip. Big was beautiful.

Frank Gehry’s design for the Guggenheim museum in Bilbao initiated the wave of cultural building that has now washed up in Dundee in the shape of Kuma’s V&A. In its wake, those commissioning architecture wanted the resultant buildings to have the “wow factor.”

For architects and clients it was very clear that size mattered. Architectural priapism was in.

It must have been difficult to push against the grain back then. Kuma did try. “Every day was difficult,” he admits. In the 1980s clients wanted exposed concrete and transparent glass boxes. “To propose wood for them was not easy at all.”

There were times when he himself wavered. At the start of the 1990s Kuma even turned his hand to the postmodern style with a wildly excessive design for a Mazda showroom (ironic given that, in 1989, he had published a book of interviews with the likes of Philip Johnson and Frank Gehry which he called Good-Bye Postmodernism).

Towards the end of the building’s construction, however, something happened that changed everything for him. One afternoon in his office the phone rang. Kuma picked it up in his left hand and put down his right hand to rest on the glass table in front of him, only for the glass to shatter, severing his nerves and exposing the bone.

When I bring the accident up he raises his right hand in front of his face. “Still, I cannot use those fingers,” he says, indicating those affected. “The nerves of three fingers was cut. I cannot feel.

“But I think it was a good accident for me. Not bad accident. Because I can learn many things from the accident. Small things. I couldn’t use the computer, so, stuff the computer.

“Before the accident I was in a sense arrogant. I could do everything fast. But after it I couldn’t achieve that kind of speed.”

From then on he committed to his idea of big is baleful. He began to use wood more and more, to speak out against monumentalism.

“Architects design big volume and big volume is very dangerous sometimes,” Kuma suggests. “Big volume can kill the environment. In the 20th century designing big volumes is very heroic. Architects are very proud of doing that. But now I feel a shame to do that kind of thing.”

You would put it that strongly? “Yeah. Architects should be as humble as possible and as careful as possible. It’s not a heroic profession at all. It’s anti-heroic.”

This is swimming against the currents of architecture over the last 30 years (at least). But maybe the tide is now moving in Kuma’s direction. In the 21st century architecture is about sustainability, about reducing the environmental impact.

The collapse of, first the bubble economy after Black Monday at the end of the 1980s and then the Japanese tsunami in 2011, which killed more than 19,000 people, and was the greatest disaster to hit the country since the Americans dropped nuclear bombs on Hiroshima and Nagasaki, has dramatically transformed the debate about architecture in his homeland.

“In Japan we changed our way of thinking. Before, most Japanese thought the concrete building was much stronger. But the tsunami changed that mentality.”

Maybe that change in attitude can be seen played out in the competition to build Tokyo’s 2020 Olympic stadium. Zaha Hadid’s practice won the original competition in 2020. The Hadid design was impressive, monumental and, in the end, unworkable.

“Most people didn’t like the shape,” suggests Kuma, “and didn’t like the budget. Almost tripled the initial budget.” (That, of course, is more often than not the story of architecture, of course, as he well knows. That said, Hadid’s original design for Tokyo had a reported £3bn price tag)

His own design, which won the second competition, was itself criticised in some quarters for its scale. “But, basically, people prefer our design. It’s very …” He searches for the appropriate word in English. “… understated. It’s a low silhouette. Most people feel sympathy to that solution.

“And, also,” he adds, smiling, “we are in budget.”

We look again at the building just across from us, the museum he has given Dundee. Tomorrow he will fly back to Japan with some duty-free whisky. He will leave a museum behind him. Scotland probably gets the best end of the deal.

KENGO KUMA: HOME AND AWAY

When he was a boy Kengo Kuma lived in an old wooden house built before the Second World War. He didn’t like it much at the time. His friends were living in new homes built after the war, all aluminium and florescent lights.

But when he started using wood in his architectural designs the perfume of his old home started to come back to him. It turned out to be the smell of the tatami floor. “To use wood reminds me,” he says. “It’s my memory of my old house.”

What is the house he lives in like now? “My wife designed my house. She is also an architect. She’s a specialist of the private house and I don’t want to fight with her. It is very comfortable. It is a small house but has a big wooden terrace. I can feel nature in the house.”

Nature in architecture and architecture in the landscape. These are the themes of Kuma’s work and life. Before he’d ever come to Dundee, Kengo Kuma would travel to Scotland to play golf. At St Andrews. At Gleneagles. He loved the links courses in particular.

“It’s very different from the courses in America. American courses sit in nature but are very artificial. Very clean designs. But Scottish courses very, very natural. And my experience in Scotland started from that experience.”

So when it comes to the V&A Dundee, the game of golf in Scotland can take some credit.

WHAT TO EXPECT IN THE NEW V&A DUNDEE

Sophie McKinlay, V&A Dundee’s Director of Programme walks us through the new museum

“Museums of the past were large and monumental, and this is much more domestic. It may not look like that when you are standing right next to it but it’s part of the city and we want to be part of the city with our programme.

“When you step into the building it’s a really warm welcome. There’s a sense of scale that is manageable for anyone walking in.

“The building just basically unfolds for you. You walk into the main hall and you know exactly what your options are. There’s a sense of informality and encouraging people to find their own path.

“I think it’s about choice. It’s about offering people multiple points of entry. It isn’t: ‘You can only experience design if you go through these doors and go to the exhibition.’

“We’ve got a project space as you come up the main staircase and that’s open to everybody. That’s an agile space where we will programme four to five shows a year. There will be young emerging artists. we will reveal hidden archives that haven’t been shown in Scotland but through a contemporary lens.

It’s a building that encourages you to feel comfortable about asking questions. It’s a non-judgemental space.”