The 1910 postcard below describes the building as 'Hong Kong Telegraph Office'.
In Soloman Bard's Voices from the past, he describes it as 'a three storey office of the Telegraph'. That suggests it was the offices of the local newspaper The Hongkong Telegraph. However, since the building was the headquarters of Cable & Wireless, I think the 'Telegraph Office' means an office where you could send and receive telegrams.
I haven't found the exact date of construction yet, but it was probably the late 1890's, like the surrounding buildings. David Derrick has a photo showing the building being demolished in 1948.