This building replaced an earlier Magistracy building a short distance away to the south-west, see: http://gwulo.com/node/16097
Completion date is a guess, based on this information from the EPD report:
The earliest magistracy on the site was constructed sometime in the 1840s, probably as part of a larger programme of works including the conversion of the Magistrate’s House into a Prison building. The building was two storeys with a central double-height court room, verandah, and various offices and waiting rooms. Throughout the 19th century the building remained in use with small outbuildings constructed to the north, but over time it became inadequate in size and facilities.
Demolition date comes form the same report:
The demolition of the old Magistracy and construction of new foundations was contracted to Messrs. Kang On & Co. at end the of January 1912 and excavation for extensive basements was started.