How to book a flight for faculty interview with no knowledge of available budget?
When you're invited for an interview, the university should offer some guidance on how to book travel and what they will reimburse.  If for some reason they don't, it's reasonable to ask how you should handle it.  By default you should book a reasonably priced economy-class ticket (i.e., given several options don't take one that stands out as much more expensive).  It's wise to save a few screenshots of price comparisons in case anyone suggests you should have gotten a much better deal.  If all the available flights seem remarkably expensive, for example if you are invited to interview with relatively little notice, then you can always check to make sure they are OK with the expense (but they will presumably say yes if the expense could not reasonably be avoided).