Bassett is a 4 to 5 hour drive from Omaha, so if you're in southern MO, I'd guess that your drive time would be more in the 8 to 10 hour range.
I suspect that flying to Grand Island would take nearly as long as driving 'cause it can't be a high volume route. Rapid City is probably a shorter, time wise, flight and probably less expensive because RC is a bigger airport.
If it was me, I'd drive, but I'd be aware that restaurants might be closed and bring a cooler with drinks, cold cuts, bread, and condiments.
St.Louis Mo. to Grand Island 1 hour 45 min.
Yes, but the OP wrote that he was flying out of Fayetteville, AR, not St. Louis, MO. I assume that since the OP indicated that he was planning to fly out of NXA, he is in southern MO.
Geographically speaking, the regional hub for NXA is more likely to be DFW than STL, but these aren't my travel plans, so I'm not going to look it up.
I've flown from Omaha to Springfield, MO, a couple of times and always connected through DFW.
