I'd start by at least checking whether the airport hotels you're considering have 24/7 shuttle service and whether it's on a scheduled basis or available when you call at around 1 am. You might even be able to arrange for the hotel shuttle to be there at the time you specify in advance. Moreover, when the hotel isn't far, a phone call upon landing can result is the shuttle being there for you, assuming that you don't need to wait for checked bags. For all you know, there could be a scheduled shuttle at just the time you need.

A limo (car service) is likely to charge the same amount regardless of distance (within reason), so that it could be a very expensive ride for what it is.