EDIT: I realise this actual is rather off-topic, more dealing with places to get various Japanese foods than the food itself. Hopefully still useful.
Just to give a bit more specific detail on particular places, and specifically the cheap places...
Generally speaking breakfast comes in 2 varieties - Japanese or "Western". I won't go into crazy detail because the first answer is awesome. Still - Japanese is miso soup, rice, pickled veg and the like. It's a pretty filling and tasty combo, though maybe not to everyone's tastes first thing in the morning! Western-style takes many forms - from 'continental'-style (breads, jams, rolls, fruit juice etc.), to coffee and donut places, to what we Brits call a 'cooked' breakfast (sausages, eggs, bacon). Some odder additions include hotdogs and hamburg-steaks (odd minihamburgers in a rather thick gracy). Try:
- Mister Donut: a coffee and donuts place. Unlimited refills, and there's pretty much one near every station
- Jonathans / Cafe Gusto / Denny's - all breeds of 'family restaurants' - not something we really do in the UK but I imagine based on the menus that they're still pretty different from anything in the States
To be honest though - breakfast is something people eat at home. Eating on the street / on the train is not common (and actually frowned upon by some), and eating breakfast at one's desk isn't common either. If you want 'real' Japanese food for breakfast I'd eat at your hotel.
Moving on... lunchtime is easy in Japan! If you're on a tight budget, there are convenience stores on pretty much every corner that sell a bunch of tasty, filling and cheap meals. Look out for the chains Lawson, Family Mart (Famima) and 7/11. Awesome-times lunch food includes:
- Onigiri - delicious filled rice balls wrapped in norii (dried seaweed) - < 110 yen
- Nikuman - a big hot doughy dumpling filled with pork - < 110 yen
- Pre-packed yakisoba (fried noodles), karage (fried chicken and rice), mabudofu (tofu in a tasty tasty sauce) - all less than 500 yen
Moving up from convenience store lunches are the Japanese-style fast food places. Again, almost always close to stations, you can get a cracking meal for well under 1000 yen. These places are also all decent options for dinner as well for some budget eats, AND for post-drinking pre-first-train hangover food scoffing.
- Matsuya - eternally popular chain selling rice bowls with a bunch of different toppings, amongst other things. You always get a bowl of soup with every meal also. Easy if you don't speak Japanese too - you buy a ticket for the food you want from a vending machine with pictures. Stock everything from gyudon (strips of beef on a bed of rice), bibindon (Korean-style rice bowl), chicken rice bowls, as well as more esoteric fair such as odd mixes of curry and rice. Everything under 1000 yen, much of it under 500.
- Yoshinoya - original beef bowl place, much the same style as Matsuya above. Simple food, low prices. Same price range as above.
- Ootoya - A little bit more upmarket, but still within the 1000-1500 yen range. More varied food in a traditional 'restaurant' setting. Usually get miso soup, rice and pickled veg if you order your 'main meal' as a 'set', so great for trying the 'standard' Japanese food.
- Ohsho - another great cheap chain restaurant. Deals in cheap and filling gyouza dishes, usually with rice or beer or other excellent staples of life. You can get a great set with tempura (veg and fish fried in light 'panko' crumbs), fried rice and plenty of tasty tasty gyouza for under 1000 yen.
Something fancier than the chains for lunch? Department stores are great restaurant hubs. Almost all will have a 'food hall' in the basement, and again almost all have a restaurant floor near the top of the building. The food halls have bentos (packed lunches) with a mix of anything and everything - veg, rice, chicken, beef, pickles, sweets, the lot. There's also a ton of free samples so if you only need a wee snack, why pay?
Back to restaurants - in the department store where I work we've got a floor with traditional Japanese places selling sushi, sashimi (actual raw fish), katsudon (fried pork cutlet mixed with egg and fried onions on rice), tempura, all kinds of noodles (thin soba, thick udon), either fried or in a soupy-broth type thing, amongst other choices (Chinese, Thai, Korean, Indian, French even). All of these places have colour photo menus, so a point and a smile should get the job done if your Japanese fails you!
One last thing - almost every department store restaurant will do a business lunch or lunch set of some kind. You can usually snap these up for less than 2000 yen.
So - on to dinner. Aside from all the places above there's still a ton of choice. You can literally walk through the area around any station and find at least 1 or 2 tiny random local restaurants, if not 10 or 20, so those you'll have to figure yourself. As for larger places...
- Izakayas! - a cross between a bar, a restaurant and a cozy den, izakayas are the go-to place for after-work food and drink for Japanese folks. Colour menus, a wide range of food (everything mentioned above as well as okonomiyaki (fried savoury pancakes), yakitori (barbequed things on sticks!), takoyaki (delicious octopus dumplingballs!) and butabara (the fattiest meat you'll ever eat!)) and almost always offering a time-limited all you can eat (tabehoudai) or all you can drink (nomehoudai) deals. Popular chains include Za Watami and Doma Doma, but where there is one izakaya there are others. In urban areas they're occasionally sorta-hidden in taller buildings, so keep an eye out for billboards outside which say what level they're on (e.g. 'Doma Doma 6F' means there's an elevator somewhere to the sixth floor, and beyond a palace of munchies!).
Should also add there's a lot of imitation-Western fast food - don't discount it all as the same as back home though. Take hamburgers - yes, there are plenty of McDonalds. But the Japanese burger chains are pretty unique and have plenty of bizarre fair. King amongst them is Mos Burger, with Lotteria a close second.
Ooft. As I think of more foods I'll add them. Enjoy Japan!