Hi there!
As your trip spans multiple JR zones, I think your choice is between a nationwide 14 day JR Pass, or individual tickets and smaller city/metro level passes. Firstly, lets get a handle on your base JR ticket costs for your trip:
-
Narita airport terminal 1 > Tokyo: ¥3140, (62mins, 79km)
-
Tokyo > Matsumoto: ¥10150, (168mins, 285km)
-
Matsumoto > Takayama: ¥11710, (304mins, 354km)
-
Takayama > Kyoto: ¥9980, (190mins, 314km)
-
Kyoto > Odawara: ¥11950, (177mins, 429km)
-
Odawara > Tokyo: ¥1450, (36mins, 83km)
-
Tokyo > Narita airport terminal 1: ¥3140, (58mins, 79km)
JR ticket cost: ¥51520 for 7 legs
- 14 day rail pass (¥45,100) saving: ¥6420
Based on this costing if you use a 14 day nationwide pass (which will cover all your areas of travel) you will make a basic saving of ¥6420, with any further JR travel being free, for example, trips to Nara/Osaka/Hiroshima from Kyoto, or use of the extensive JR network around Tokyo (JR Tokyo map) will all be free. Because of this I recommend using a 14 day JR pass for your travel.
A 14 day rail pass will not provide 100% coverage as you will need to use non-JR transport to access some of your destinations:
Hakone
The JR Pass will get you as far as Odawara for Hakone. From here you should use the non-JR Odakyu operated rail and tram system to get up into the Hakone area. I recommend purchasing the Hakone Free Pass on arrival into Odawara.
Snow monkey onsen at Yudanaka/Jikokudani
For this you'll need to use the JR Pass to get up to Nagano (1hr from Matsumoto) and then use non-JR Nagano electric rail to Yudanaka (sample itinerary from Matsumoto). The total trip will be around 2 hours. How about staying overnight in Yudanaka onsen rather than Matsumoto?
Matsumoto to Takayama
While you can get the JR train (via Nagoya) that connects these 2 locations, you could also consider using the non-JR Nohi bus to do the trip in 1/2 the time and travel through the alps. Here's the [relevant timetable].
Shirakawago
Shirakawago is only accessible by non-JR buses and cars. Nohi bus operates a good service from Takayama for this. You can find more information on the [Nohi bus schedule and prices to Shirakawago here].
Metro systems and travel within cities
While the JR Pass will cover all the main travel between cities, and is very useful within Tokyo, it is not accepted on non-JR metro and bus systems in Tokyo, Kyoto and Osaka. For these cities you will need to look into citywide passes, or at purchasing single tickets as you go:
You can purchase these tickets/passes at the relevant station before travel, or by visiting the tourist counter at the JR station you arrive into.
Hope this helps!