Tokyo is by no means a bicycle-friendly city. Bike lanes are almost nonexistent and you'll see no-parking signs for bicycles everywhere. (Ignore these at your peril: your bike could get impounded, requiring a half-day excursion to the pound and a ¥3000 fee.) Despite all this you'll see locals on bikes everywhere.
Cogi Cogi is a bike-sharing system with ports around the city, including some hostels. There are instructions in English, but it's a little complicated to use. You'll need to download an app, register a credit card and have wi-fi connection on the go to sync with the ports.
Some accommodation has bikes to lend, sometimes for free or for a small fee. For exploring the neighbourhoods of Ueno and Yanesen, check out Tokyobike Rentals Yanaka. In Yurakuchō, Muji has bikes to rent.