Regarding sumo, the tournaments run for 15 days from Sunday to Sunday, and the July tournament is held in Nagoya (Aichi prefecture), about 3 hours away from Tokyo by bullet train (or 3.5 if you head there on your way back from Hiroshima). This year’s dates are the 7th to the 21st. The last day usually sells out, and sometimes weekend days are crowded, but most weekdays don’t require advance purchase of tickets. Without adv purchase, you won’t be close, but the Nagoya stadium is small. It’s also very hot in southern Japan in July, and more so in the stadium, especially considering the seating is pretty cramped, so I’d recommend picking up a fan from a convenience store or dollar store somewhere along the way (a small towel might also be a good idea). I enjoy sumo, but even so I had a crapton of fun attending the tournament, so much that I had to go to another one, so I highly suggest you make it if at all possible.
As for other things you might want to do, here’s some Tokyo stuff. The monorail to Odaiba island gets you to the Future Museum, which has that space-suit looking bipedal robot, and I believe the life-size Gundam statue is still up there. I know they took it down from the original exhibition a few years ago and then they put it back up, but someone would have to confirm that it’s still up for you. On the same island you can also visit Tokyo Sea Life Park, which isn’t the hugest aquarium in the world but they had some cool exhibits like penguins.
In Tokyo proper as Machine Elf mentioned there’s the Tsukiji Fish Market, which if you’re into the whole ridiculously early morning thing you can take a taxi (no trains that early) at around 4 AM to see the fish auction. It’s very cool just to watch people doing daily business. Registration for entry starts at 5, but people start getting there kinda early to line up, so arriving by 4:30 might be safest. There’s a map on this page that shows you where to go, and you can see where that is on this google maps page (just zoom in a bit and you’ll see the curved structure from the other map).
Even if you don’t make it early enough for the auction there’s some excellent sushi at the shops in there. The most famous one is easy to find; just walk around until you see a line of people waiting 90 minutes to get in the door. The shops anywhere there are gonna be great however, if you don’t feel like waiting. There’s a small gardens just down the street from the market area, which if you look on that google map you can see just to the southwest. It’s within walking distance, and is a very nice place to stroll around while you digest your sushi.
In Asakusa there’s the famous Kaminari Gate at the Senso Temple (Sensō-ji - Wikipedia). Both gate and temple are beautiful, and there’s a nice set of shopping stalls all along the path from gate to temple. There are even people offering rickshaw rides on the street outside the gate there if that’s your thing. There’s also a small [drum museum](Welcome Asakusa Taiko Drum Museum - Tokyo) in the area that’s kinda neat.
I remember liking the Sapporo Beer Museum in Ebisu (Shibuya). It has stuff like old Sapporo ads, the brewing process, and a seating area to buy some different Sapporo brews that aren’t sold anywhere else. There are also a lot of museums in Ueno, like the Tokyo National Museum. There’s a zoo there as well that’s supposed to be pretty good, though I haven’t been.
Other areas that might be fun to just walk around a bit are Harajuku (and the Meiji shrine), and Ginza, which has some higher-end shopping, including department stores selling Japanese makeup that a friend swears by and can’t find anywhere outside the country. It’s fun to go by night to Shinjuku (which has a 7-story sex shop) and Shibuya (the place where Hachiko waited for his owner to return, or went to get treats from the food stalls there, pick whichever version you like), which has a vaguely famous super busy intersection right outside the station.
For general stuff I’d say you should hit up karaoke somewhere. Even if you don’t think you’ll enjoy it much you might be surprised, and it’s pretty cheap if you just pay for the time (like 700 yen/person/hour). You get your own private room, and there are tons of English songs to be found. An hour might slip by quicker than you think once you both start belting out Build Me Up Buttercup! If you’re into it, bar culture there can be fun, and there are enough people who speak English in Tokyo that you might get some friendly locals trying to strike up a conversation with you (although I’d avoid Roppongi if you’re looking for that sort of nightlife).
As to foods see if you can find horse sashimi (called basashi), which tastes pretty well like a regular meat to me, and makes me wonder why we don’t eat it in more places. Also if you like pickles (oshinko) they pickle just about everything over there, so if Oz is anything like the States get 'em before you’re once again confined to pretty much only pickled cucumbers again. Fish is great in general even if it’s not raw, and I’d recommend a type of mackerel that’s very popular grilled there, called hokke (hoh-keh). There’s a picture of how it’s usually served on that wiki page, and you can find it in most pub/restaurants (which are called izakaya(s)). Oh, if you do make it to Nagoya for sumo, the city (or maybe prefecture) specialty is an eel dish (hitsumabushi), which was excellent, but grilled eel anywhere you can find it is also very good. Cold soba (buckwheat noodles) dipped in broth is great in the summer, and if you haven’t had Japanese-style ramen you haven’t lived.
) About the only other specific experience I want to do is to attend a Sumo match.
