Mir area. But Yaffa works good for me. So Beis Yisroel/ geula/ Yaffa...
The public library at Givat Hatachmoshet (next to the public schools across the street from the UN compound) might be your closest option but their hours are strange (Sun, Mon, Wed, Thurs 2pm-7pm and Tues 9am-1pm). They only have 5 computers or so and last I checked they charge 5 or 10 shekel an hour (it's been a couple years, but I've heard it's still up and running).
Another option is the national library on the Hebre U campus at Givat Ram (not Har Hatzofim), you can take the 68 from Bar Ilan. They have dozens of computers and it's free.
Another paid place as of a few years ago was the convenience store next to cheese and tomato (a few doors down from the great synagogue). He had 2 computers and it wasn't cheap but conveniently located.
There also used to be one in the Tachana Merkazit called Cafe Net with like 25 computers (2nd floor I think, 12 shekel an hour).
Also, others that I've heard of:
Pelephone store at 2 Moshe Luntz (across from Cream 'N Dream & Cafe Rimon), Ben Yehuda Area. 10 shekel an hour.
Cafe Internet, 31 Yaffo Street, 12 NIS per hour.