Winter Osaka - Nagoya - Takayama - Kyoto - Osaka. What Pass?
I had drafted my itinerary as below:
Day 1 (18th Jan)
Day 2 (19th Jan)
Day 3 (20th Jan)
Day 4 (21st Jan)
Day 5 (22nd Jan)
Day 6 (23rd Jan)
Day 7 (24th Jan)
Day 8 (25th Jan)
Day 9 (26th Jan)
Day 10 (27th Jan)
Day 11 (28th Jan)
I think the first step is to work out what your JR travel costs could be. Here they are in as much detail as I think is needed:
JR ticket cost: ¥52,780 for 13 legs
As you can see, you'd make good savings with a 14 day Japan Rail Pass (this is the only JR pass that will cover all of your legs), and would also have the flexibility to do any further JR travel for free. However, if you decide not to take the trip to Hiroshima, then I think you would be better off simply purchasing JR tickets as you go (and using the 1 day metro passes in Kyoto and Osaka).
For Takayama & Shirakawago, the JR Pass will get you to Takayama on the fantastic Wide-View train, however you will have to use the non-JR Nohi bus service from Takayama to Shirakawago (¥4300 round trip). It is very easy to travel to Takayama as it is not a big town, and Shirakawago is also very small. Personally I recommend saving money and doing it using the JR Pass + Nohi bus. You can also consult this great free guides from the Japanese National Tourism Organisation for more information on how to tour Takayama and Shirakawago effectively.
Hope this helps!
Order your Japan Rail Pass Online
Starting from $252.00 for 7 days of unlimited Japan Rail travel