Is it okay if I want to buy a drink for the girl sitting next to me on the plane?
It sounds reasonable, Why do you want to buy that woman a drink?
I want to show her how I can be a nice person and also make a good impression so she can become my girlfriend.
While it is okay to buy drinks for a person that you are interested in, she does not have the ability to leave her seat and it'll create a very uncomfortable situation if your romantic advances aren't welcomed.
Shit, I was hoping to get into a relationship from the flight.
Start with small talk and if both of you are enjoying the conversation, offer to buy a drink when they trolley passes. And if you don't hit it off, you are only out a few dollars but if you handle the conversation well, you may have gained a new friend.