The cheapest way of eating while travelling in Japan? Leave it to me, I am an expert on this, trust me =D
First, land in the southern Japan, NB!
Avoid Big Cities.
Buy a big cheap 30 kg rice bag from a storehouse and ask your couch surfing host for permission to use their rice-cooker.
Don't leave the house and eat rice for 30 days. Yeah baby!
Just kidding
It really depends on why you are going to japan and where but...
I have some
exclusive personally developed tips for you concerning
food:
1. If you wanna eat cheaply/healthily/tastefully cook food at the place you are staying at whenever possible, provided that you can cook well using cheaper ingredients and you know what and how to use the stuff that's on sale in Japanese grocery stores. ( if you haven't been to Japan before, I highly doubt that you can orient well in a supermarket, even if your Japanese is near-native. Will take time to get used to it.). In this case, you might want to buy a bento-box (good ones for 500-800 yen) or a few and cook some stuff in advance for the day and take the stuff with you. (if you choose this option). Frozen vegetables are sometimes cheaper than fresh ones. Chinese cabbage is very cheap (probably). And moyashi is also cheap!
2. Eat out as little as you can, it's obviously the most expensive choice.
However it's an important part of culture so...
let's say you might wanna ( in other words definitely should

) consider this option too.
Don't really expect to get by cheaper than 800 yen in most places. I doubt that you're gonna search for the cheapest places where to eat at for the whole 30 days, so be ready to spend at least 1000 yen on average each time you eat out (probably more)
3. The no1. aka. the best single never-mentioned , aka
Free option:
Meet 1-3 friendly new Japanese people who are a bit older than you every day and go to eat out with them ( however you're gonna do that). There is a 50-80% chance that they will treat you the first time they meet you (speaking from experience) I don't know how old you are though. But anyway, almost every new Japanese person I went to eat out with for the first time treated me. Especially when you meet several people.
4. Go to parties/nomikais at Japanese people's houses (I mean, go, if they invite you).
5. The last one but not the least one:
The best option if you don't have much time, are lazy and want to grab something fast ( not the cheapest or most interesting option though):
Go to a Convenience store ( they are literally everywhere, like a network of I don't know what). If you use GPS, search for the nearest convenience store on the map( provided that you know what their logos look like).
Walk to the shelf where they sell bread (パン)
and look at the calorie count of each product and roughly calculate the best available amount of calories per your money and buy those breads. Bread is the cheapest and healthiest option of the cheapest ones, I think. There are also cheap chips or something that are pretty stomach filling ( like Taco chips). Look at the calorie count and choose your product. If you haven't been to Japan before, you probably won't mind eating all those breads anyway, as they will all be probably a new experience to you. Still, I'd say that it's not as healthy as real food (cooked by yourself or bought when eating out) and nor is it as tasty.
But well, you wanted to know the cheapest ways...
Use the advice at your own risk and on your own responsibility.
And, never be shy to slightly mention that you're on budget in any situation. There is a good chance that people will be eager to help you out with advice or perhaps with something else better than that.
However don't be persistent or mention it too much. Just let people slightly know that you're on budget mmmkay? And ask questions.
+
make 100 yen shop one of the first places you're gonna go to. It is gonna save you a lot of money if you'll understand what kind of stuff you can buy there.
Food's gonna cost a considerable amount of money too, so you might as well consider on cutting down on money spent on food as well, not only transport.
And as for the cheapest transport: Hitchhiking. ( Never tried it. Perhaps I'm going to try it out though).