Vega,
Good call on not climbing Mt Fuji this time of year. It's way too friggin hot. June is the best time if you can find a day when it's not raining.
Recommend Ginza district and a walk through the Sony building. The emperor's grounds should also be on your list. Got to save a night for Roppongi.
There is a nice Italian place called Soho. Very very good. It's in Harajuku(
http://www.soho-s.co.jp/). English and Japanese website. You will need reservations.
PM me if you want any more specific info on Tokyo. Lived just South of Tokyo for 3 years.
Bdger