It's opening on 1 May 1914 was announced in item 126 of the Hong Kong Government Gazette 1914 (Supplement):
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GENERAL POST OFFICE. No. S. 126.-- It is hereby notified that a new District Post Office, to be known as The present "Western Branch" will in future be known as the "Sheung Wan Branch Post Office" E. D. C. WOLFE, |
It wasn't a new building, it just occupied the southern / uphill end of an existing Sanitary Board building. The Post Office section of the building is shown on this map:
and between the two vertical lines marked on the photo below:
I believe the post office closed on 25 Apr 1964, see https://gwulo.com/media/54978
The post office relocated to temporary premises further up the hill, while this site was re-developed.