Hi,
Just to give a few pieces of info. First, you can buy it before going to Japan or once inside, and unlike some other JR regional passes though there is no discount for getting it before you get on the plane. You can in fact buy the rail pass at Fukuoka Airport at the TISCO Travel Information Center (on the 1st Floor of Fukuoka Airport, International Terminal) but there is no JR train between the airport and Hakata Stn. Be sure to read the complete info on rules, conditions, etc here. You will need to purchase a subway ticket to go to Hakata Station, but it is not expensive and takes only about 5 minutes.
You can buy both the 5 and 3 day passes, and have them start as you want. Just remember that once it is done there are no changes allowed, so make sure your schedule is set in concrete. For the Yufuin no Mori reservations, any JR office in the service area can do it.
It is quite easy to recoup the costs of the pass, but if you want to check, run your itinerary through Hyperdia.
For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide.
For Fukuoka, don't miss the Nanzoin Temple which is one of Japan's finest yet almost totally unknown to the outside world and completely free!
Look into a Fukuoka Tourist City Pass also.
You can also see a lot of previews on the best there is to see here.
In terms of history and varied points of interest, Nagasaki is really the most interesting city on the island. It is worth a couple days to see the city, and another day to see areas nearby like Shimabara/Unzen or whatever interests you.
For Nagasaki, the best sights are the Peace Park/Museum - but if you see the one on Hiroshima you might skip it - Glover Garden, Dejima, and Koshibyo Shrine. Gunkanjima would also be well worth seeing, but it takes 3 hours. And at night be sure to go see Mt. Inasa at night for a spectacular night view over the city.
Kumamoto has some excellent places to see as well. One of the best hot spring areas on Kyushu is at Kurokawa, which unfortunately you'd need to take a bus at least part way there, but offers a great forested outdoor bath and serene environment. The castle is also one of the most famous, and be sure to go visit Suizenji Garden which is one of Japan's most beautiful.
Mount Aso is also a spectacular live volcano, but
has been closed near the crater lately due to higher poisonous gas emissions.
Oita is famous for its hot springs as well, such as in Yufuin as well as Beppu has plenty but also its Hells Tour and nearby Takasakiyama monkey park and aquarium are both terrific.
And if you still have time, Kitakyushu has a few good places to explore. Kokura castle is a very nice one, plus its Moji Retro Area will take you back in time and Sarakurayama has one of Japan's finest mountaintop views of the city and is nearly unknown to the rest of the world.
Please remember that with a JR Kyushu Pass you cannot take the bullet train between Hakata and Kokura.
Good luck.