A tune-up? What's the dealer going to do; check tire pressures, fluid levels? Seems like a waste of money to have some dude do that for $shop rate/hour when that can be done at the airport when you get back to your car.
One month is no big deal.... it's not like it's got carbs and the float bowls might be empty after a month.
Sounds like it's a business trip your going on.... I'd take a cab to the airport and expense it rather then having it sit in the economy lot at the airport. If you do get someone to start it up every two weeks, be sure they run it for at least 10 minutes at varying engine speeds. Just letting it idle for 3 minutes will actually kill the battery even more since the alternator rarely provides much of a charge at idle.