It depends on a lot of the situation. A larger airport, with established staff, will be different from a small town airport hiring someone to be the only permanent staff.
Identify the key stakeholders and their issues. Then see how that maps to what the airport manager will have the power or ability to address. That should lead to some good questions.
If, for example, an issue is lack of hangers, ask what kinds of approaches he or she would take to that problem. Don't expect one answer that will be perfect, and don't argue with the candidate (or the committee) about the proposed solution, but probe his or her answers for how creative he or she is, how many options they know of, if they have resources (like other airport managers) to get ideas from, and see if in general you like his or her approach to addressing a problem.
If the answer is "here's how we will do it" and there is only one way, and you get the impression it is the only way he or she wil consider doing it, you may have a flexibility problem.
if the answer is entirely a "run it like a business" answer, that may or may not be the right answer for your situation.
If instead, the manager proposes a variety of solutions, and talks about how they may or may not work in your situation, and shows the ability to adapt, and that they have researched your situation some, that would be good, IMHO.
Likewise for other issues.
As far as airport-specific issues, you might want to find a successful airport of similar size/situation, and ask their manager and/or commission for an hour of their time to ask them what they consider important issues and questions when hiring...
I'm really curious... if you are in a position to do so, can you tell us more about this?
--david