you're brave, but you'll be okay! Most of the airplanes I've been on have a changing table in the restroom, though I'd bring like a throwaway towel or something to lay down. As for airports, some have "family bathrooms." I remember some larger, and more international airports even having a nursery - crib, seats, changing table, sink, etc. But honestly, we've changed our boy's diapers just about anywhere - right in the waiting area, in our airplane seats.
I've also known some parents to rely on benadryl, or to "tire their kids out" beforehand so that they sleep for a good portion of the flight. But, I've seen that backfire as well. Good luck!