IDJ has uploaded several photos and notes about this. They're linked to below.
I've set the completion date to December 1962, based on this entry in the Chamber of Commerce yearbook for 1962:
The Sea Terminal owned and operated by The Hongkong and Kowloon Wharf and 'Godown Company Limited, opened during December 1962 to accommodate the "Himalaya", and is the first tangible evidence of the Company's policy to provide the Port with reception facilities of an international standard for passengers and transient tourists arriving by sea. The new Sea Terminal will be replaced in three years' time by the completion of an Ocean Terminal, construction of which is due to begin shortly. The basic design for this major project is a pier 1 ,250 feet long by 250 feet wide. It will accommodate either two modern cargo vessels of about 550 feet each or one luxury pasenger or cruise liner on either side. Depth of the harbour will limit the pier's capacity to 35 ft. draft vessels. The pier will be a three storey structure, the first level providing cargo working space and the upper floors public and passenger concourses, restaurants, shopping arcades, a super-market, roof-level car parking, and many other modern features.
The interim Sea Terminal provides a fully air-conditioned Concourse, 300 feet in length, housing seventeen selected shops, Restaurant and Bar, Cable, Telephone, Postal, Information, Tour and comprehensive Baggage services.
I've set it's closure date to 1965 based on the three year life mentioned above.
UPDATE JAN 2020: I've changed the demolition date from 1965 to 1969, as the photos below from Klaus shows it was still standing then.]