Yemen Net/Yemen Mobile is the only provider that I am familiar with. They offer dial up and DSL. According to what we were told, 24 hour DSL access is only available for businesses, but this may be inaccurate information so ask when you go.
All over the place. Around 1-2 riyals per minute. Some of them have wireless internet capabilities and most I have been to have DSL, although I have run into some with dial up/slow connections.
Most remain open during Ramadan and some may close for a bit to break the fast, but other than that, they seem to be open as usual during Ramadan.
A few of them:
Internet Kingdom (the Diary across from City Mart). They have a printer.
Lamasat. The Diary, north of Imaratul Khowlani. They have a color printer and scanner.
White Net. Shariah Taiz, north of Shariah Shumaylah. They have a printer.
Compute Me (in the vicinity of Shariah Haddah about 1 block east of Haddah, north of Shariah 45. I have never been but it seems pretty big.
Again, you can find many on smaller streets, but we prefer going to some of the nicer bigger ones for comfort and safety, insha Allah