Rangers set up an Ibrox William Hill Scottish Cup quarter-final against Albion Rovers with a 4-0 win over Dunfermline on Friday, the team they lead by 23 points at the top of League One.
Ally McCoist's team also have a Ramsdens Cup final against Raith Rovers to look forward to on April 6 and McCulloch, who recently claimed Rangers deserved more credit for their record this season, is desperate for any silverware.
"It's a good chance to put a cup run together," said McCulloch, who was promoting the club's quarter season tickets.
"We are in another cup final, that a lot of people have forgotten about, and we are doing very well in the league, if you are looking at the stats.
"We have a chance as players, management, to have a very, very successful season.
"Scottish Cup, league, Ramsdens Cup, any success you have got to grab with two hands.
"When you are playing at a club like this, you are judged on success and what you win. You are not judged on the amount of appearances you have. It's all about 'What have you won'?"
McCulloch recently argued there was a "very thin line" between League One and the Scottish Premiership and he feels the cup action, with only three top-flight teams having booked their place in the last eight so far, has vindicated him.
"Yes, it does," he said. "I think that was blown out of proportion slightly.
"What I was saying was that there wasn't as big a gap as people were making out from some of the SPL teams to the so-called lower-league teams.
"I didn't insinuate anything about Celtic. What I meant was maybe the bottom six of the SPL."
And McCulloch feels League Cup finalists Aberdeen should be considered favourites ahead of Rangers, despite the Ibrox club's bigger budget.
"You've got to look at Aberdeen, already in a cup final and doing really, really well this season. So, for me, you would probably say Aberdeen. We know what we have got to do. We have a chance to progress and that's all it is.
"I'm delighted to get a home tie but it's going to be a difficult game.
"They've got a good manager and a decent team and no doubt they will make it difficult for us. It's a game we definitely don't think we have won already."