There are a few people on here who live in Kyoto i think. I used to live in Kobe for many years, but never cycled to Tokyo. Can't offer any advice on routes, but just for Japan generally (and i did some touring in summer in other parts)
It is fine to camp anywhere, so bringing tent/bivy can be useful. Summer will be hhuummiiiddd, so be prepared to drink crazy amounts of water. If you aren't looking to camp, then staying in Popeye internet cafes is your best bet.. 1000yen or so for a reclining chair, your own booth, free shower and ice cream. The cheapest way to stay in Japan for sure, and certainly adds to the experience. Otherwise you may be able to find some 'rider houses', which are hostels mainly designed for motorbikers, but usually open to cyclists, and are also around 1000yen for a bed for the night.