My experience is that it depends entirely on the airline and, sometimes, on the reps working the desk that day.
Generally speaking, my understanding is that you are obligated to give what they take you as a standby. However, once you're okayed to fly as standby, I think you're then working under the same rules as everyone else.
As long as there are first class seats available, they'll generally let those on the upgrade standby list pay the extra cash and sit there. But if there are more people wanting to upgrade than they have available first class, most airlines have a priority pecking order: they'll start with their "preferred" customers, and go down from there until they get down to a who-asked-first basis. Depending on the airline (and I don't know about AA's policy), your standby ticket may or may not put you at the bottom of the pecking order for upgrades.
You could always try to call and find out (if you speak other languages fluently, that is.)
Otherwise, you're right to just be happy when you find yourself walking down the jetway! Get there early.