you can get day trips from tokyo into the mountains and i think they go as far as where the monkeys are. i wanted to do this when i went and had it booked but it was cancelled due to a typhoon!!
did just over a week in tokyo and 4 days in kyoto which is definitely worth going to - completely different to toyko with lots of temples and gardens. you can get between the 2 on the bullet train - about 2.5 hrs and about £60 one way, but worth it. if its a clear day you can see mt fuji out of the window.
for things to do in toyko buy a good travel guide and explore using the underground. its really easy and all the different areas of the city are quite different. harajuku, shinjuku are good to see also the shrine at asakusa - theres a street 5 mins walk from there thats full of shops selling japanese tea pots and cups,knives and sake pots at decent prices.
also in toyko - the hama-rikyu garden is worth going to-it has a ceremonial tea hut in the middle of the lake which and was impressive cos it was a really old traditional garden but surrounded on every side by glass skyscrapers
youre going at cherry blossom time aswell i think which is one of the best times of the year to go
from the airport to tokyo - get the airport limousine bus - its basically a coach to the main hotels in tokyo but is the easiest and cheapestvway to get to/from the airport