Hi,
Yes you can, and to make a seat reservation you need to go to a JR Station to request it. But regarding the airport, JR does not have any line to it - only the city subway. But it's easy to take a 5 minute ride to Hakata Station. You will need your passport with a Temporary Visitor stamp to qualify. Note that you cannot take the bullet train between Hakata and Kokura Stations with it.
If you have time, seeing Fukuoka is a great place. 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.
For Beppu, its hot springs are famous but there are also other things to enjoy, such as the Hells Tour and nearby monkey/park aquarium. If you are interested in hot springs, then visiting Yufuin is also a good idea, but one of the best on Kyushu is Kurokawa in Kumamoto.