Yes, you can use the rail pass on the bullet trains (except nozomi class bullets, which are 10-15mins faster on the Tokyo-Kyoto route as they stop at slightly less stations).
You'll need to spend over ¥28,300 on individual rail tickets to make a 7 day rail rail pass worth while. Lets take a look at your costs:
- Narita airport terminal 1 > Kyoto: ¥15530, (240mins, 592km)
- Kyoto > Tokyo: ¥13020, (161mins, 513km)
- Tokyo > Nagano: ¥7770, (105mins, 222km)
TOTAL: ¥36320 for 3 legs
Based on these costs the 7 day rail pass looks like a good option - you'll save around ¥8020 over standard tickets.
By the way, as you intend to travel on the 1st of January, make sure you reserve your ticket to Nagano well in advance. Japan Rail Pass holders can reserve JR-East network bullet trains online for free here. The Tokyo-Nagano route is valid on this system. Please make sure you pick up your reserved seat ticket as soon as you get to Japan to avoid being charged for your reservation (you must pick up your ticket up to 1 day in advance of the date of travel to avoid a charge).
Hope this helps!