I think you should be able to find hostels in some places in Japan for around US$30 per night. You will have to stay in the same room with backpackers or other people. If you are really cheap, some Japanese people sleep in manga cafes or even in 24 hour fast food restaurants, but I don't know if they will let a foreigner get away with that.
As far as the food, the user who said you can check convenience stores gave good advice. Also, maybe check out the Tokyo Vegetarian site here