It's not a question of needing the pass - it is a question of if you save money with it. With your itinerary, you do better than breaking even, so it is worth getting. You cannot use the pass on city subways and buses though - it is of limited use in Kyoto, to go see the Fushimi Inari Shrine as well as to go to Arashiyama. In Tokyo you can use it on the Yamanote Line, which goes to many popular places.
The question of getting to Mt Fuji is a more difficult one. On the mountain itself there is very little to see or do, so most would end up going to Hakone or the Fuji 5 Lakes area. You cannot climb it in December anyway. Both offer something a bit different. It is easier to get to Kyoto from Hakone though, by taking the bullet train from Mishima. You cannot use the pass to get around inside either place though. Look into a regional pass for each place, like a Hakone Free Pass.
In Tokyo a few of the best places to see are the Tsukiji Fish Market, plus the site that every last tourist goes to see, the Sensoji Temple. If you are in Tokyo on a Sunday afternoon, be sure to go visit Harajuku to see the youth with their wild fashions. Meiji Shrine and Yoyogi Park are next to it and well worth seeing also. And be sure not to miss the Shibuya Hachiko intersection especially on a weekend when it looks like half the city is crossing the street at once.
In Kyoto you could spend weeks there and not see everything. The three best sights though are the Kinkakuji Temple, Kiyomizudera Temple, and Fushimi Inari Shrine. Fushimi is the most time consuming; you could spend a couple of hours to over half a day if you want to romp over all the trails.
Nara is often neglected by some tourists, which is a shame. Missing the Todaiji Great Buddha in Nara is like going to India and skipping the Taj Mahal. Nara Park has a lot of great places. A few other great places missed out even by those who go to Nara though are the Isuien and Yoshikien Gardens. They are gorgeous and not crowded at all.
For getting to/from Narita airport, use the Narita Express to maximize the value of your pass.