You can do Yakushima Island as an overnight trip, if you are willing to take the tour bus. I would suggest that your look over the Yakushima Monkey website for more information:
http://www.yakumonkey.com/Here is the basic plan:
(1) Get the jetfoil and Kotsu tour bus (A & B) tickets in advance in Kagoshima. This can be done at the tour offices in Kagoshima-Chou Shinkansen Station.
(2) Board the 7:45 AM Jetfoil to Yakushima. Arrive at 9:45 AM.
(3) load luggage in tour bus. Tour A starts at 10:00 AM.
(4) When tour A ends, the Tour bus will drop you and your luggage off at your hotel (usually the JR or Green Hotel).
(5) Tour bus picks you up at your hotel around 7:30-8:00AM, and you start tour B.
(6) Dropped off at 13:10 (1:10 PM), 20 minutes before the Jetfoil leaves port (1:30 PM). Jetfoil arrives back in Kagoshima at 15:30 (3:30 PM).
From here, I was able to get back to Kagoshima-Chou station, and catch a limited express to Miyazaki. I wish I had spent three nights in Miyazaki before traveling up the eastern coast to Beppu.
From Miyazaki, I traveled to Nobeoka, and caught a bus to Takachiho for a night. The next day, I took a bus back to Nobeoka, and continued up the coast line to Beppu. I was able to go on the afternoon tour bus of the "Hells of Beppu". I would spend an extra day in Beppu, and possibly visit the monky park and the Aquarium, along with an afternoon trip to Yufuin.
There are a lot of other places in Kyushu to visit. Fukuoka, Nagasaki, Kumamoto, and Kagoshima come to mind, but there are also smaller places like Arita, Mount Aso, and Kurokawa to see.
Good Luck on your trip.