I like hanging out in Fukuoka. I think I spent only 3 hours at my hotel last time through. Besides the sites in the city itself, there are day trips of interest to Yanagawa, Daizaifu, and other places depending on your interests.If you haven't been to Hiroshima, you might want to go to Nagasaki instead of spending time in Fukuoka.
As for the outdoors, the Aso area is probably the easiest to get to. You would take a relay-tsubame train to Kumamoto, and then a train/bus combination to the crater area. You could hike around the crater or to the top of Takadake, or down to the caldera floor.
Kirishima NP on the Kagoshima/Miyazaki border is a better hiking area but would take a bit longer to get to. It would be another train/bus combo to get there. A lot of hikes start or end out of the Ebino Kogen area.
Kyushu Rail Pass is only available to foreign tourists.
http://www.japan-guide.com/e/e2361_05.html