You didn’t actually say what time of year you are going so some activities outside of Tokyo may vary depending on season.
Hiking Mt Fuji is a great experience but you really need to chose your time carefully. Go at the wrong time and it is a 5 hour queue of people walking up and you can get very frustrated. Dawn from the top is pretty spectacular though and the run down the Osunabashiri on the Gotemba side is fantastic - basically a long lava gravel run.
Harajuku, Meiji Shrine, Shibuya, Shinjuku, Akihabara, Tsukiji are all great places to go. Harajuku is best around Sunday lunchtime. Having sushi at the outer Tsujiki market is amazing but might spoil your taste for any other sushi afterwards because it is so good. Asakura temple is a popular destination and you can take a river boat from there down to Odaiba to visit the Toyoda Car showroom, Venusfort shopping mall and Oedo Onsen and the Museum of Maritime Sciences (looks like a big concrete boat!) before taking the Monorail or walking back across the Rainbow bridge.
Warning, if you are going to go to an Onsen, Tattoos need to be covered at all times. Tattoos are associated with Yakuza and are not acceptable in general public anywhere in Japan.
Depending on the time of the year, there can be lots of little festivals around which are always fun to participate in.
One of my favourite places is Asakura Choso Museum. It was the private home of one of Japan’s best sculptors and there is plenty on display but for me, the beauty of the place is the garden and the house itself. http://tokyoreview.blogspot.com/2007/02/museum-asakura-choso-museum-yanaka.html It is a easy to find if you know how and there is a wonderful community shopping street on the way so if you are interested I will send you some instructions.
Getting out of Tokyo is also easy and worthwhile. Nikko and Kamakura are only a few hours away by train and have amazing temples and Shrines. In case you didn’t know the difference, Temples are Buddhist, Shrines are Shinto but they often share the same grounds. Japanese people don’t feel the need to only have one religion!
One of my favourite whole day trips is to Hakone. Train from Tokyo to Yamato-Hakone then Mountain train, Funicular, Cable car, ropeway, pirate ship, bus. Beautiful area. Make sure you stop for eggs cooked in a Volcano and visit an onsen. You can stay in a Ryokan if you want to make it a two day trip and if so, going to the Hakone Open Air Museum is very highly recommended. More information here - http://www.japan-guide.com/e/e5210.html, http://www.kuriositas.com/2010/07/hakone-japans-amazing-open-air-museum.html
If you go in ski season, you can go to a number of ski fields easily from Tokyo - none easier than Gala Yazawa which has it’s own Shinkansen station and ski hire and gondola stop all in the same building. You can get from your hotel to the ski field in 2 hours.
I lived in Tokyo for 5 years and have so many things I miss about the place. I will back there myself in October for a couple of weeks so have plenty of things that I am planning to do when I am next there.