I'd probably ask to be dropped off at Jackson Square - that's the heart of the FQ where the cathedral is and plenty of artist and musicians gather around the square, so it is a lively and fun area. Then I think you can comfortably spend about 3 hours before getting a taxi back to the airport - head out at about 5, get to the airport at about 5:30, which is plenty of time before a 7pm flight.
Regarding timing, it's about a 30 min taxi ride between the airport and the French Quarter, so if you land at 1, you should be able to be there around 2. I'd take a taxi to the French Quarter - wander around, have something to eat.
