Tokyo Town Hall. Governmental building in west of Shibuja. Two towers, one can use lifts in each tower to go 45th floor and enjoy the sight, for free. On clear days you can see the Fuji. Also nice to visit the townhall in the evening.
Akihabara: town district, “electric town”. Electronics everywhere.
Ikebukuro: district north/west, nice stores with electric gadgets
visit an Onsen, bath with water from the earth. Excellent for rainy days. I.e. the one at Odaiba. No bathdress or towel needed at this Onsen. http://www.ooedoonsen.jp/
groups and organizations in Tokyo
if you are linux-enthousiast/opensource fan: visit https://tlug.jp - great people around, english is spoken
Amanohashidate fishers village, mountain with lift
Ine, Kyoto prefecture: port seeing/ship cruise
trains: Japan has some special trains, many very expensive though. Twilight express, train suite, ななつ星in九州. Tickets start at $2.500 for one overnight trip.