First thing you should do is check out which matsuri coincide with your trip to Japan. You should visit some if you can.
Plan your days so that on national holidays and friday nights / weekends you're in places that you'll enjoy people watching (Akihabara, Yoyogi, harajuku etc) and not in places where you'll be stuck in long lines waiting to get in somewhere (Tokyo tower, Sky tree etc).
Do you know Hiragana/katakana? If you don't then it will be nice if you spend a little time learning them. It will help you on your trip, and it will make your trip much better.
Don't ignore the food, there's so much different and lovely Japanese food that you could (if you wanted) never eat the same dish twice while you're there. Japanese food (unlike for example: the food in some other western countries like the UK) is a HUGE part of their culture, don't be affraid of it. You'll miss out if you do.
There's some places that could be considered standard for any tourist, for example:
Shibuya crossing/Hachiko (btw, seen the movie?)/getting lost in the surrounding areas.
Yoyogi Park.
Ueno Park.
Shinjuku.
Harajuku.
Akihabara.
Ikebukuro.
Maybe Odaiba too.
BUT what you do in Tokyo largely depends on your interests, and you can do only so much in Akihabara. for example:
Depending on how HUGE of an anime fan you are you may want to visit more places like: nakano broadway, studio ghibli museum (needs advance reservations), or maybe go to comiket or Tokyo International Anime Fair (that's if your dates coincide with any of those) etc.
If you like / don't mind J-pop you may want to
visit akb48's Theater and the idol training cafe in Akihabara.
If you like Jazz you may want to visit
the pit inn. Also, there's a huge punk-rock, noise etc underground music scene (much of it plays live in places in Shimokitazawa as far as i know).
If you appreciate old retro video-games there's several places that you could buy em in Akihabara and there's a lot of places where you could play em.
Maybe you're big on fashion and should go for some shoping in Shibuya.
Maybe you enjoy art, there's various art spaces and there are art some fairs in town throughout the year.
Maybe you enjoy history so you'll want to visit
47 ronin graveyard (Seen any of the movies about them?) or
Yasukuni shrine & the war museum.
Maybe you're a fan of
the Japanese martial arts or enjoy watching
Kyudo etc, it won't be easy finding a way to see some of that - requires serious planning.
Sumo, do you like it? Would you spend some money and time on it? It's not on every month. When will you be going btw?
Maybe you enjoy the folklore: Youkai, kaidan? (Seen
the movie Kwaidan by Kobayashi? Seen
gegegenokitaro anime?) Museums? (Especially Edo-Tokyo museum you shouldn't miss) Traditional theatre like:
Kabuki and
Noh? (
Kabuki-za is opening soon) etc etc etc.
Tokyo is a huge lovely chaos of a city, and doesn't have so much your typical tourist sites. In my opinion the point of visiting Tokyo is to enjoy it like you would if you lived in it.
Kyoto is the exact opposite, it's a town that has many very specific sites and areas worth visiting.
You should also consider possible side-trips. For example: Kamakura, Yokohama (which has a great Chinatown), Nikko, Fuji, Enoshima.
The following links could be helpful to you:
http://nhk.or.jp/nhkworld
begin japanology videos on Dailymotion
japan-guide.com - Japan Travel and Living Guide