My experience in Tokyo is from ten years ago when I was a student with limited cash reserves, so you won't find any high end cuisine here, but that being the case, here are my thoughts.
I would visit some of the Tokyo neighborhoods - the area around the Asakusa temple is a good one. Or around Ueno. Go to a sushi restaurant during the day, sit at the bar, and just point things out. Or ask for a chirashi sushi bowl and let the sushi chef do his magic.
Conveyor belt sushi is also fun and relatively cheap without the pressure of having to explain what you want.
Yakitori-ya are nice as well. The standard is several small skewers of grilled chicken, but they also serve dozens of other foods - from odd little mystery meat balls, to chicken skin or cartilage, to all assorment of vegetables. A fun casual meal you can find all over the place.
Almost every restaurant has plastic food in the window showing what they serve. Feel free to drag the waiter to the window and point out what you want. There is one street that only sells kitchen supplies, including the plastic food if you want some unique souveniers. The street is referred to as Kappabashi. (The street really has another name). Somewhere around Ueno station if memory serves me.
Another cheap favorite are the Okonomiyaki-ya. Okonomiyaki is a kind of omelet mixed with vegetables and meat that somehow does not sound so appetizing by my description but is really quite wonderful. There are specialty restaurants that only serve Okonomiyaki - hundreds of them. ("ya" means store in Japanese, so sushi-ya is a sushi restaurant, yakitori-ya is a yakitori restaurant. Etc.)
And of course there are hundreds of options in Roppongi - the big night life area - or around the massive shinjuku or shibuya stations. I even had one memorable all you can drink beer, all you can eat mutton meal on the roof garden of a department store near Tokyo station. I never know I could eat so much mutton. There is just no end to what you can ge tin Tokyo.
I wish I were going with you. There are 80,000 restaurants in greater Tokyo and I'd like to try them all.
Also, the publisher Kodansha has an excellent bilingual Tokyo atlas with extensive detailed maps, english translations, and the best Tokyo subway/train maps you will find anywhere. Japanese train stations do not always have roman alphabet translations of the station, so unless you know the kana, it can be easy to get lost. The atlas can solve all those problems. Here is the Amazon link.
http://www.amazon.ca/Tokyo-City-Atlas-B ... 4770028091
Living in Tokyo for 18 months, this was the most valuable thing I purchased.
For a fun cheap day outside, go to Yoygi park on a Sunday (near Harajaku station) and see all the crazy Japanese kids out in their weekend costumes.
For what its worth, horse sushi (thinly sliced raw horse meat), a traditional Kyoto specialty, isn't all its made out to be. Go to Kyoto for the temples, not the raw horse meat.
And stay away from Natto. That stuff is nasty.