Depends on the airline and it depends if you are in their Frequent Flyer program. It will vary but for instance, American Airlines will give you the finger if you ask to upgrade to First Class if you don't have status (at least last time I tried years ago before I had platinum status), other airlines will charge full price while others will only charge a reduced rate. I was on a Delta flight a few years ago and got a first class seat for 50 bucks at the counter (not a member of any of their programs, got delayed on other airline so missed my original flight, hence why I was on Delta). So bottom line, best you can do is ask at the ticket counter hope it's not sold out or they want to charge you hundreds extra to fly in first class.
Anymore, unless you are flying more then two hours, I really don't even find it worth it unless it's a packed flight, you get either the bulkhead or exit row and all you give up is maybe a meal and a bit more elbow room.