I live in Tokyo.
For a very nice group to join that has different ride levels that are all a little "touristy" try:
http://halffastcycling.com/ For a bike rental: this will be a challenge. If you read Japanese, there are companies that will deliver a rental complete with clipless pedals etc to your hotel and pick it up. If you don't read Japanese, you may want to ask the hotel if they can help you. Nothing in english.
For locations:
Oi Futo is great, but a little tricky to find, and only good riding on Sundays when the trucks are not there.
Tama is too narrow, don't recommend.
Arakawa river is awesome, but you'll need to ride about 15km to get there on the street. Riding on Tokyo streets is easy, the cars are polite and the paving is excellent. But you'll have to stop at red lights.
Imperial Palace is possible, but unless you go at 5am, its gets trafficky. You're expected to stop at red lights, even though there's not really any reason to. But of course, best to respect the local cycling culture.
If you're not familiar with Tokyo and don't have another cyclist to bring you around, I'd suggest just running. Imperial palace, akasaka palace, yoyogi park are all good places to go. Swimming is getting tough because pools are closing for renovations for the 2020 olympics. But you can still find a few public 50m pools open...for now...