It was built in 1818 by architect William Burn, who pioneered the Scottish Baronial style.
He was best known for his work on country houses such as Blairquhan Castle in Ayrshire.
The Long Room, which will host the museum celebrating George Wyllie's work, is a replica of a Sir Christopher Wren design in London.
The Custom House was occupied by Her Majesty's Revenue and Customs until 2010 when it ended its operations in the building.
A £1.6 million refurbishment of the east wing of Custom House began last year.
The works are being carried out by Riverside Inverclyde, an urban regeneration company working on a stretch of the waterfront between Greenock and Port Glasgow.
The building lies some 400 yards from Greenock Ocean Terminal and opposite Greenock Central Station.
It also sits next to the contemporary-style, £10m Beacon Arts Centre, which opened in January this year.