I've flown with my now 7-month old son 3 times (round trips) - at 2 months, 4.5 months, and 7 months. Each flight has been easy. Babies at this age want to be held more than anything, so it's actually easier than a long car ride where you can't hold her. Make sure you breastfeed/bottle feed her on the ascent and descent. Basically, if you swallow to pop your ears, make sure she is swallowing too. You should be able to pre-board with her, which makes getting on and off easier.
I recommend bringing a sling to carry her in so that you can have your hand/hands free to carry other stuff. I have changed my son in my lap (put the pad down on your lap first) - he finds it tremendously amusing, although he is so long now that it's hard. Some larger airplanes may have flip-down changing tables in the lavatories, but don't count on it. Better to just board with a fresh diaper. The flight is short. Bring a couple of books and toys and you should be set.
If you buy her a ticket you can take on her carseat. Honestly, for a baby this age I don't think it's worth the trouble, especially when flying alone. Where are you going to put her while you try and install the thing? I would check her carseat instead. Airlines are responsible if they get lost. If for some reason it doesn't show up on that flight's baggage they can give you a loner car seat and take it back once they deliver yours at your destination. (I just learned this a couple weeks ago - I had been prepared to wait in the airport another 5 hours for the next plane to show up!)
Kids at this age travel really well. Good luck!