The Intercontinental and Sofitel are within a 10 minute cab ride from FAA International airport and are nice for one night. Papette is not really the touristy part of French Polynesia, most of the islands fit that touristy profile more accurately.
I can't recall if there are any hotels super close to the airport?
On a side note, French Polynesia and the word "cheap" don't really go in the same sentence.