Hello,
The JR Pass covers all the places you wish to see, including travel to Beppu and Fukuoka. I personally love Kyushu and visiting that part of Japan is a great plan. I would also recommend a visit to Nagasaki if you have a day free.
The best way to travel between Fukuoka (Hakata station) and Beppu using the JR is to take a train from either place to Kokura and from there travel on. You'll be coming from Hiroshima, so basically the route would be Hiroshima -> Kokura - Beppu - Kokura - Fukuoka.
Lastly let's have a look at ticket prices and see if the JR Pass would be useful for you.
Narita Airport - Tokyo ¥ 3,220
Tokyo - Kyoto ¥ 13,800
Kyoto - Osaka (return) ¥ 1,120
Kyoto - Hiroshima ¥ 11,290
Hiroshima - Beppu ¥ 11,630
Beppu - Fukuoka (Hakata) ¥ 7,730
For a total of ¥ 48,850 the price of a 14 day JR Pass is ¥ 46,390 so far you would be making ¥ 2,460 in savings. Remember that the pass is also usable for local travel in Tokyo and Osaka on the local JR lines. So I certainly think that a JR Pass is worth considering, also since it offers so much flexibility and you can take any train you like for 14 days.
Hope this helps!
Daniel
