Japan isn't the most accommodating place when it comes to those of us who don't eat anything and everything. If you're a vegetarian of some stripe, or you have allergies, or there are simply things you don't like, then you may be apprehensive about eating in Japan. Here's some information gained by a very picky eater after six months in Japan.
Eggs and dairy are actually fairly common these days, contrary to an out of date perception. In the traditional Japanese restaurants, they have two cheese-based dishes on the menu. One is rice-cake and melted cheese and the other is eggplant and melted cheese. These restaurants are not great for vegetarians; while you can get a reasonable meal at them, it would soon get repetitive. They offer salads (green salad, mixed salad, tomato and onion, cabbage and miso) and a range of seasonal vegetables that you grill yourself at the table on a table-top charcoal barbecue. But that's about it for pure vegetarians. Much of what these places serve is sashimi and yakitori (skewered grilled meat).
Noodle restaurants are good places for vegetarians. Many are specific to either udon (white wheat noodles) or soba (dark buckwheat noodles), though some places do both types. Both are available hot or cold and in a soup base or not. The soup bases vary, and while some are fish stock based, you'd have to be sensitive (or allergic) to fish taste to notice it. They will typically have vegetarian options such as mushroom noodle soup or "age-dashi dofu," deep fried tofu pieces in soup, and may do tempura as well. Some places offer "yasai" or vegetable-only tempura - but many places do only mixed tempura, which is vegetables, fish, and seafood. If you go with someone who likes fish and seafood, you can swap veggies and fish/seafood around. If you ask for vegetable-only tempura, they will usually accommodate you even if it's not on the menu.
Chinese restaurants usually have a range of tofu and mushroom-based dishes as well as seafood and fish dishes, so you should be fine there. Ethnic cuisines are often altered to local tastes and ingredient availability, so it's worth trying non-Japanese restaurants.
There are many Italian restaurants and they're fine for vegetarians, with pasta and pizza in vegetarian and seafood variants.
Some restaurants are very accommodating - the okonomiyaki restaurant in Queens Square Mall near the convention site doesn't have a vegetarian item, but they were happy to provide one based on the vegetarian ingredients that they used.
I can also recommend the following book: "What's What in Japanese Restaurants" by Robb Satterwhite ISBN: 4-7700-2086-4 ($11.95)