Hey Chris,
What people do on visits is usually a matter of taste. I'll throw in a couple of offbeat suggestions.
But first, you're too late to climb Fuji-san. It's already closed to all but pro climbing teams who have to get permits for off- season climbs. You'll probably get a great view from the Shinkansen on the way too and from Tokyo though.
Tokyo - the area where most of the equipment bought by restaurants is called Kappabashi-Dori. The public are welcome, and you can buy plastic sushi, fake beers which look drinkable, etc. Up the street and just down a side road is a knife shop called Tsubaya, the finest collection of handmade cooking knives I have every seen. A large percentage of sushi chefs buy their knives there. If you like to cook, this area is well worth the visit.
If you're in Tokyo on a weekend, go to Harajuku station and try both the shops and the park. You'll see a lot of counterculture going on, it's great fun.
An offbeat place to shop, with lots of crafts and second- hand places as well is in Shimo-Kitazawa. Lots of cafes and places to eat, too.
For nightlife, Roppongi is famous and justly so. Try a Robotoyaki restaurant - Izakaya is expensive but great fun and well- worth the cost.
Personally, I would skip Ginza for shopping - all the high- end stuff you can get anywhere.
Lastly, I'll leave Osaka to others who know it better for comment. (I would spend one more day in Kyoto!) There is a JR limited express train which goes from Kyoto to the airport and makes only a couple of stops (including, of course, Osaka.)
Have a great trip.
