Hi Again :D
Now that is going to be a great trip! Personally I really love Kyushu and this trip already looks amazing.
Sorry I could not follow up with you Yesterday, it was getting a bit late here in Japan but here we go!
I was looking at what pass would help you out best and it was a bit hard to find something that would fit ideally.
I think the [14 Day JR Pass] will be your best option. You can use it for all your train travel including to and from Tokyo and Narita Airport.
To see if it would be cost efficient I made a little overview of normal ticket prices:
Narita --> Tokyo: ¥ 2,940
Tokyo -->Kyoto ¥ 13,220
Kyoto --> Fukuoka (station name is Hakata) ¥ 15,210
Fukuoka --> Kumamoto ¥ 4,990
Kumamoto - Aso ¥ 2,180
Aso --> Kurokawa By bus 960 ¥ but not covered by JRPass :(
Kurokawa Onsen --> Oita to Beppu by bus(2.300 Yen) and on to Beppu by JR ¥270
Oita Yufuin (return) ¥4,020
Oita --> Kagoshima ¥ 15,540
Kagoshima --> Fukuoka :¥ 10,170
Fukuoka --> Tokyo ¥ 21,720
Tokyo --> Narita :¥ 2,940
For a total of ¥ 93,200 the 14 Day JR-Pass is ¥ 45.100 saving you at least ¥ 48100
For Kagoshima, I would recommend walking up in Shiroyama park to see a breathtaking view of the city, ocean and Sakura Jima an active volcano in the bay of Kagoshima. You can visit it too, by ferry. The Senganen Gardens are very beautiful and peaceful and may be worth a visit too!
Takachiho can be a bit hard to visit as trains no longer go there. I would recommend reading the Wikitravel page about it, as has a great list of additional options of how to get there.
Hope this helps!
-Daniel