cheap food
I know this is an old thread, but could still be helpful to people going to Japan.
As far as cheap food goes, the best I can tell you is this:
When I was in Tokyo in 2013, I was on a tight budget, and was very un-familiar with Japanese food. So I stuck with American fast food chains and convenience stores. As far as the fast food goes, the prices are only slightly higher than in the USA (& better quality then in the USA!). Plus you have a pretty good idea of what you are getting. Fast food menus are similar to USA menus and most places have menus available in English. Convenience stores are everywhere in Tokyo. More 7-11s and Lawsons than I've ever seen in the USA, no joke!! As with any country, regular restaurants are a definite budget buster. Yes, you should try some local restaurants, it's part of the cultural experience. But American style fast food chains & convenience stores are cheap and easy. Plus, whenever you are out & about in Tokyo & get hungry, there is almost always a convenience store or fast food joint near by.