Our son, now 2 1/2, flies at least once a quarter (since age 6 months) and now has miles accumulated for free tickets on 2 airlines.
One comment on Iowajes' point about oxygen masks: before anyone panics, of course every seat has oxygen masks. the concern is an *extra* oxygen mask, for babies riding on laps. That is commonly on one side of the plane, on smaller aircraft. (right-hand side for Delta, not sure if it's consistent) It is something to ask about if you are not reserving a separate seat for your child.
We did hold our son sometimes on short flights, but generally we bought him a seat, and brought his car seat onboard. You cannot hold your baby through turbulence. (look up the definition of "moderate" turbulence) and a wrap-type carrier cannot, either. For us, it was about safety more than convenience, but I have seen all these strategies employed by people.
Of course, with a carseat the stroller becomes the carseat and/or stuff carrier. We did buy a heavy duty bag for gate checking it, and it worked well.
At his age now, we use a Cares Kids Fly Safe harness. He is still getting used to is--does not sleep as readily, as he is more upright. But, sitting in the full seat means he can't kick the seat in front of him, and gives us some extra room, sitting next to him.
Finally, another option you might look into if you can't borrow a pack-n-play is to rent one. Many resort areas have supporting industries of baby equipment rentals. Maybe there is one near your friend. Also, if you bought one, do you visit regularly? It can be more than a one-time item. (in our case, Grandma bought one for grandchild #1, and it has been reused for quite some time--great purchase!)