If you're going to be flying into Eureka Springs Northwest Regional Arkansas Airport  (XNA) is 50 miles away from Eureka Springs. Rent a car from the airport when you get in and then drive into town. Little Rock is 4 hour drive. Don't do a taxi or shuttle from XNA unless you are willing to drop nearly 75 to 100 bucks. A car is best.  Springfield, MO is two hour drive. Private airports large enough and ones that have facilities for your gulfstream or lear are in Bentonville. Again about 50 miles.  Carroll County Airport in Berryville, 10 miles away has no facilities to speak of.