Japan North Alps - beautiful but often crowded, plus a bit expensive to get to from Tokyo. Huts are rather expensive.
Japan Middle alps - not bad, easy access to peaks via a cable-car/ropeway
Japan southern alps - Not as pretty as north alps but cheaper to get to, assuming there is actually a bus going to the particular part you want as not so popular. Huts cheaper but you get what you pay for...
Mount Fuji - not much point unless you want the 'cachet' of saying you climbed the highest... Basically a big pile of gravel.
Yakushima - very far from Tokyo, must fly really and dont think there are any direct flights. Nice place though, and all the huts in the mountains are free. Can get really busy in peak season. Hokkaido is probably less busy but similarly far from Tokyo (but in opposite direction).
Basically you can find all this kind of stuff in your Lonely planet guide
Generally wild camping is frowned upon/illegal in national parks unless there are no huts - the only National park that I think you are allowed to wild camp in is up in Hokkaido - and that is really wild - think brown bears etc.
Best place for outdoors gear in Japan is probably Kanda/Jimbocho area of Tokyo - pretty much can get anything there though probably cost more than you are used to. Other than that check out a Montbell store - these are spread out around tokyo and other cities.
Eating out is generally quite reasonable/cheap in japan, but accomodation is often somewhat expensive. Transport varies - taxis are expensive but buses trains are quite ok - especially within cities (the intercity Shinkansen 'bullet' trains are rather expensive, though you can often get a local train on the same line for less than half the price but taking many more hours). UK drivers should have no problem adapting to Japan driving conditions, but not sure how easy it is to hire a car for foreigners (this is purely from a language barrier point of view, there is no beaurocratic reason why you cant hire one). Speed limits are quite low but traffic can be quite busy in the cities - I wouldn't bother hiring a car in Tokyo. All the 'motorways' are toll roads - and the tolls are pretty damn high, but if you want to get anywhere in a reasonable amount of time then you have to grit your teeth and pay them.
Summer is really hot over here but not so bad up high (pretty bad in the cities though). Winter is mild and dry on east side (e.g. Tokyo) but involves metres of snow on west side. Snow shoes are essential for many 'hikes' during this period. Best time of year is probably spring and autumn, though can get typhoons coming from late summer to early autumn. Rainy season is around June in Tokyo so another reason to avoid summer.
Actually, if you have some specific questions then you can PM me - I may be able to help, or know someone who can.
