Champagne Court Block A at the back behind the camera shop has an internet cafe, the computers are not very private however! They also get packed in the evenings.
Last time I was there they had some slightly more private (but more expensive booths) near the counter area.
But tbh I use that place all the time (or used to) to write reports and nobody batted an eyelid seeing as 90%+ of the men at Champs court are punters themselves
The internet cafe staff don't give a damn about it.
Alternatively there are numerous internet cafes in Chungking Mansions, however as you may expect there is absolutely NO privacy as the computers are packed together super tight and it is easy to look over each others shoulders.
The MTR internet terminals dotted about do not let you look at 141.
Nor do HK Libraries either!
There are three in Mongkok, not East Mongkok Mongkok., Take the exit to Ladies Market and MK cyber centre (freewifi here 141 friendly) and there is a small doorway leading upstairs which is a game cafe which is pretty much a netcafe. 141 friendly nobody gives a damn.