cheapest 'hotels' in Japan would definately be the capsule ones. You can get some for about 2000 yen a night. Most are men only though but you can check.
I used a Japanese website to book my hotel and it was pretty cheap about 6000 yen a night and there were still cheaper ones on there (this was for Kyoto). try here
Rakuten Travel
As per food, have something light in the morning, liquid thruout the day and something filling at night.
Get the JR pass if seeing many different cities, definately worth it