the biggest airport in the state is in little rock. there are smaller commercial airports around the state, in fort smith, for instance, but i am not sure where else. several cities have small noncommercial airports for charters or personal planes. nearby large airports in neighboring states are tulsa, oklahoma, and memphis, tennessee.
you can get a shuttle bus between the little rock airport and the memphis airport
("bette's bus," 800/752-8745; also try 501/821-7797 or 501/223-3338 for limousine service) and, i think, between the little rock airport and downtown hot springs (501/821-7797 or 501/223-3338 for limousine service). better check on these for sure before you count on them. i am guessing that the airport management office for little rock (501/372-3439) is probably the best place to start looking for this information.