Is shooting pool Japan popular?

Yeah there are many places. In Japan they call it ビリヤード　biriyaado (obviously from the english 'billiards') but it's actually just American 9-ball pool not the original billiards.

There are pool-halls all over the place, I've only ever been in the evening after drinks but I guess they'd be open in the day as well. It seems quite a popular evening activity, although not as popular as karaoke etc. Actually there are sometimes pool tables in sports-bars and large karaoke complexes, but they usually prefer to put in dart boards. I suggest you look for them in the popular night-life areas, i.e around bars/karaoke places etc. If you're living in Tokyo I wouldn't think you'll have any problem finding one.

I've seen a few snooker tables in some of the pool clubs but not that many; if you search around you might find some. Also if you're planning to play a lot it would probably be a good idea to join one of the places.. you'll probably get a membership card and some discount every time you play.