There is an airport in Tunica:
http://www.tunicaairport.com
They don't have any regular major carrier air service yet though, but they do have quite a few private jets that come in and out. I thought I heard on the news that Delta was going to begin offering some sort of flight from Atlanta to Tunica, not sure where the details on that stand though.
The bottom line is you basicly have to fly to Memphis and then get to Tunica, a rental car is your best bet.
You know years ago when they first proposed buiding the Casinos here the companies wanted to build them in Memphis, but the Church going holy rollers threw a stink and said it would increase crime in Memphis - one of the most dangerous cities in America.
So what you need to do is fly to Memphis, go straight to Tunica and when you are ready to go home, go back to the Memphis airport. I would only use Memphis for that, it is just too dangerous to go shopping or sightseeing in Memphis, you should spend as little time in Memphis as possible.