Hi Veronica,
This trip looks excellent - you'll cover a lot of great sights in Japan with it.
I think that a 21 day will give you the best savings as you will be doing expensive JR travel from day 3 up until you return to the airport. That said, here are your rail pass options and the costs associated with them and your itinerary breakdown:
14 & 21 day rail pass costs
14 day: ¥45,100 / ¥3221 per day
21 day: ¥57,700 / ¥2747 per day (+¥12,600 over 14 day pass)
Itinerary breakdown
- Narita airport terminal 1 > Tokyo: ¥2740, (62mins, 79km)
- Tokyo > Nikko: ¥5230, (121mins, 150km)
- Nikko > Tokyo: ¥5230, (117mins, 150km)
- Tokyo > Kamakura: ¥1700, (48mins, 51km)
- Kamakura > Tokyo: ¥890, (58mins, 53km)
- Tokyo > Kyoto: ¥13020, (164mins, 513km)
- Kyoto > Nara: ¥690, (45mins, 41km)
- Nara > Osaka: ¥780, (55mins, 48km)
- Osaka > Kiitanabe: ¥5090, (146mins, 175km)
- Shingu > Hiroshima: ¥17340, (380mins, 759km)
- Hiroshima > Hakata: ¥8500, (69mins, 280km)
- Hakata > Kumamoto: ¥4790, (35mins, 118km)
- Kumamoto > Aso(kumamoto): ¥1080, (99mins, 49km)
- Beppu > Tokyo: ¥23550, (441mins, 1228km)
- Tokyo > Narita airport terminal 1: ¥2740, (58mins, 79km)
TOTAL: ¥93370 for 15 legs
As you can see you stand to make an excellent saving with the 21 day pass (a whopping ¥35,670) so I recommend this option for cost and also for general ease of travel. There are a few exceptions where I recommend using local bus services in your itinerary over JR:
Kiitanabe > Hongu > Shingu. While it is possible to use JR to loop back to Shingu from Kiitanabe, I think you're meaning to traverse the penninsular. As such, I recommend you look into public buses from Kiitanabe to Hongu and then onto Shingu and have omitted these legs from your JR pricing.
Mt Aso > Kurokawa > Beppu. Again, it's possible here to loop back from Aso to Beppu, but I think you mean to travel up to Beppu stopping at Kurokawa on the way. As such I recommend you look into a bus from Aso to Kurokawa, and from there onto Beppu and have also omitted these legs from your JR pricing.
You are rather busy towards the end of your trip. How about considering the Mt Aso experience in between overnights at Fukuoka and Kurokawa (Hakata/Fukuoka(night) > Aso > Kurokawa(night)), and skipping Kumamoto (unless you have a reason to go)? This would give you an extra 2 nights to play with. I'd also look into whether you can visit Kiitanabe, hongu and shingu at the same time as a day trip from either Kyoto or Osaka or with only a single night stay to further free up time and allow some more 2 night stays in your itinerary.
You can look up train timetables and costs for all of your legs at the Hyperdia travel planning website and learn how to use Hyperdia for Japan Rail Pass holders through our video blog.
Hope this helps!