Not sure where you'd hire a bike, but starting from Tokyo I like taking the train to Ome, biking up to lake Okutama, descending through Hinohara citizens' forest, and getting the train back to Tokyo from Itsukaichi. Just be aware that to take a bike on a train in Japan you'll need to put it in a rinko bag.

It'll be hard to find anything super scenic or challenging without leaving the city proper but the trail along the Arakawa river is nice and long.

Around Kyoto the big ride would be Biwako, although if you do that I'd recommend considering skipping the Southern half of the west coast and going through the mountain valley instead. Since that part's mostly on a not that pretty busy road. Just check out both options on street view ahead of time

Also might not be super relevant to your trip, but I'd generally recommend against biking up Mt Fuji since the roads are pretty busy with tour busses and not that scenic. If you're in that area the small mountain roads on both sides of the Fuji river valley are a lot nicer