For the amount of time that you are going to be in Kyushu, it would be better to get either an All-Kyushu SunQ bus pass, or an All Kyushu JR Kyushu Pass. There are plusses and minuses to each pass.
The SunQ pass is less than the train pass. It will allow you some local bus and ferry transportation in additional to long distance transportation. It is also slower, and your time in Kyushu is limited. Even though it is slower, it also has overnight bus transportation, so your daylight time of visiting Kyushu may be more.
There are also places that you have to take a bus to get to, such as Takachiho, and several of the hot springs resorts. So even if you get a JR train pass, you will need to pay for a bus to get you there.
The question that you first need to ask is, what do I want to see and do? Here are some suggestions of what you may want to do:
Fukuoka is a very lively city at night, and is well known for it Yatai food stalls.
http://www.japan-guide.com/e/e2161.htmlNagasaki in a very international city, with lots for see and do.
http://www.japan-guide.com/e/e2162.htmlThere is a theme park called Huis Ten Bosch, wich is known for it's floral displays.
http://www.japan-guide.com/e/e4477.htmlBeppu is built on top of the worlds second largest volcanic caldera, and has lots of public hot springs.
http://www.japan-guide.com/e/e4700.htmlIf you are into ceramics, you should check out Arita.
http://www.japan-guide.com/e/e8300.htmlMount Aso is an active Volcano, along with Sakurajima in Kagoshima.
http://www.japan-guide.com/e/e4552.htmlhttp://www.japan-guide.com/e/e4601.htmlMiyazaki is an older honeymoon resort area, and has a few special shrines dedicated to newlyweds nearby.
http://www.japan-guide.com/e/e8000.htmlTakachiho has a fantastic Gorge, and a nightly dance re-enactment at the shrine.
http://www.japan-guide.com/e/e8050.htmlThere are also excellent small hot springs towns, such as Yufuin, and Kurokawa, Ibusuki, and Unzen.
http://www.japan-guide.com/e/e4750.htmlhttp://www.japan-guide.com/e/e4575.htmlhttp://www.japan-guide.com/e/e4677.htmlhttp://www.japan-guide.com/e/e4453.htmlOh, you probably don't have time to squeeze this in, but there is also Yakushima Island.
http://www.japan-guide.com/e/e4650.html Without knowing your likes and dislikes, I cannot help you pick the ideal trip to Kyushu. I would suggest that you look through the links that I have given you, and decide what interests you the most.
I would say you best bet, time wise would be to stay in Fukuoka the first night, and visit Canal City, and the Yatai food stalls that evening. I would then get a 3-day pass to get around, and which pass would depend on where you end up going.
The second day is up to you.
The third day is up to you.
The forth day will be getting to Kagoshima, and if you have time there, I would consider taking the ferry over to Sakurajima (covered by the SunQ bus pass, if you go that route), before boarding your flight.
Good luck with your trip.
