Hi there!
For your trip the best Japan Rail Pass option looks to be a 14 day rail pass. As a 14 day pass costs ¥45,100, you'll need to aim to spend more than this in JR tickets for it to be an economical choice. Lets take a look at your base JR costs:
- Hakata > Beppu: ¥6930, (104mins, 188km)
- Beppu > Tokyo: ¥23550, (441mins, 1228km)
- Tokyo > Otsuki: ¥2660, (86mins, 87km)
- Otsuki > Osaka: ¥15200, (279mins, 604km)
- Osaka > Kyoto: ¥540, (29mins, 42km)
- Kyoto > Takayama: ¥9380, (193mins, 314km)
- Takayama > Beppu: ¥21480, (494mins, 1029km)
TOTAL: ¥79740 for 7 legs
As you can see with a 14 day rail pass you make a saving of ¥34,640 over normal JR tickets. You also have the added flexibility to mix in day trips if you'd like (for example to Nikko or Hiroshima). Please note that although I didn't include it in your ticket calculations you can use the JR Pass extensively around Tokyo on JR lines. Here is a map of JR lines in Tokyo for reference. For Mt Fuji I assumed you will visit Kawaguchiko. The JR Pass will take you as far as Otsuki, from where you will have to change to non-JR services for the final leg up to Kawaguchiko.
For your trip you'll be able to use all bullet trains except the Nozomi and Mizuho services, so for the large legs up and down from Tokyo you'll need to use either Sakura, Hikari or Kodama bullet trains with a change at Shin-Osaka. These services are very fast and simply stop at a few more stations than the Nozomi and Mizuho services en route.
A final comment - as you can see the trip up and down to Tokyo/Takayama from Beppu is quite time consuming. You could consider stopping off en route to Tokyo (perhaps at Osaka/Kyoto/Hiroshima) to split your journey up on both the way up and way down. Also, I suspect Takayama will be a better choice directly after Mt Fuji as you will have less far to travel.
You can look up train timetables and costs at the Hyperdia travel planning website and learn how to use Hyperdia for Japan Rail Pass holders through our video blog.
Hope this helps!
