I've moved the Naoshima journey to the second part of your trip, as it's much easier to get to and from Osaka/Kyoto. Lets have a look at the possible legs and costs you may have for your trip:
- Narita (guessing here, if it's Haneda this will be cheaper) > Tokyo - ¥2,940
- Tokyo > Nikko (return) ¥10,860
- Tokyo > Mt Fuji (return to odawara - see http://goo.gl/Y6qiz) ¥6,880
- Tokyo > Kamakura (return) ¥1,780
- Tokyo Yamanote line ¥500
The total cost of all legs comes in at ¥63,040. If you are going to do all this in 9 days, then a 14 day Japan Rail Pass would be well worth it. However, it's quite a busy schedule to fit into 9 days, and you'll be doing an awful lot of travelling.
I'd recommend you think about reducing your itinerary here. Perhaps spend a bit more time in Tokyo initially, then activate a 7 day pass for the traveling down to Kansai Airport. You could consider doing a few less day trips from Tokyo, and maybe cut out the Naoshima journey (though awesome, you really should stay overnight for the full experience, and that means 2 days).
Not only will you have more chance to soak up the atmosphere, and travel at a slightly slower pace, it will mean that you will save a lot of money just using a 7 day instead of a 14. Money that you could use on a nice meal, or a hotel upgrade.
Having said all that, if you still want to get to Naoshima, I wrote another post using the Japan Rail Pass to get to Naoshima that may help.
Anyway, the choice is yours! Hope this helps!