The closest airport is Fayetteville, AR (XNA) and is an hour drive to Eureka Springs.
Eureka Springs is a 6 hour drive from Dallas or Austin.
Eureka Springs is an 18 hour drive from Phoenix.
If flying from Phoenix, Tulsa and Oklahoma City might be an option. The drive from Tulsa to Eureka Springs is about 3 hours. The drive from Oklahoma City is 4.5 hours.