Sadly, it's not possible to bed down right in Madurai Airport (IXM). However, you'll surely find a great place if you venture further afield.
Unfortunately, there are no accommodations on the doorstep of the tarmac. But if you don't mind taking a short road trip, the Kadambavanam Ethnic Village Resort is an excellent option. You'll be in good hands here, but make sure you leave enough time to get to the airport.