There are quite a few public operated indoor pools in Tokyo. And if you don't mind paying more for the private-run ones, most major hotels have pools.
Swim wear and caps are usually available for sale at the pools, but it's best to bring your own since you never know when they'll run out of stock.
Note that in the very center of the city there are more expensive hotels and less reasonable-prised public pools, while in the residential area you will find more cheaper pools.
Public pools tend to be more particular about being clean and healthy (at a lot of pools, everyone is to get out of the water every 30 min. or so).
Other small policies differ depending on the pool. Some might have special lanes for skilled swimmers, others may have certain hours for them.
The following is just an example of a Japanese list I found from a very quick search. These are all indoor pools, and if you scroll down, the left hand yellow block represents each ward (ku). You will see the price at the very right. For example the top one is 600yen (400yen for ward residents)/per 2 hours.
Btw, it's too late for outdoor pools.
Have a nice stay!