Arriving Fukuoka airport is awesome!! Unlike Narita (that is a painfully far from Tokyo downtown) or other regional airport, Fukuoka airport is smack in the middle of downtown (sort of). Even closer to the city-center than Haneda,Tokyo.
The transportation in/out of Fukuoka airport is extremely fast!!
Oh, how I wish the discoutinued route from USA to Fukuoka comes back someday....(I guess not.)
Everything the other posters said above is true. If you have 2 weeks, you can explorer so much of Kyushu island, and maybe up to Hiroshima/Osaka/Kyoto area.
Mid-May is a fantastic time to travel. You are lucky! that timing is awesome.
Enjoy!