Try the Convention Center -- free
There's a large Trade Development Council library on the ground floor of the waterfront convention center in Wan Chai. Just come out of the Star Ferry and make a U-turn. Dozens of free computers, all you have to do is go to the information desk and sign up for a time slot. Most of the time there is no waiting, and you don't have to use the computers for research purposes. There has been free internet here sinced they finished completion of the convention center in 1997.