There are questions I'd try to ask him in a relaxed private setting, and then there's the ones that aren't a waste of time in a public forum. Anything that smacks of throwing certain players under the bus, even by omission in talking about others, will be answered in ways that are not useful. "Which players do you most look forward to working with?" would be an example. Ditto for disparagement of his predecessor, and certainly of the present FO top to bottom (or his prior employer for that matter). And he's not going to divulge whatever secret sauce he may feel he has for running a cohesive clubhouse.
Still, gotta try to glean something useful from the new guy. How about:
Compared to other managers, what do you delegate, and where are you most hands-on?
What's the biggest change fans will notice about the team, apart from W/L you hope for?
What are your thoughts on batter platooning, and conversely L/R matchups for your pitchers?
What's the most important thing you learned as a manager the first time around?
I dunno, maybe I'll think of some others.