An ability targeted by Good Stiff Drink, can be used a 2nd time. All costs must be paid and conditions must be met like normal.
The ability must also still be in play to be used.
If you use a dude's ability and experience them, the original ability is no longer in play, and isn't a legal choice for GSD.
If you use a dude's ability, choose it with GSD, and experience them before using it, it is no longer in play to be used.
To answer your edit:
You don't choose a dude.
You choose an ability on a Dude, that has already been used.
Only that ability can be used a second time.
If you experience a dude, his new ability is not that ability.
Shootout: Attach a Hex to Jia Mein (paying all costs) to make him a stud.
Jian Mein exp
Noon: Play a Condition Noon action, reducing its cost by 2, to make Jia a stud.
Noon: Boot a spell on Jia and pay 1 ghost rock to take a Condition from your discard pile into your hand.
Jia Mein is in a shootout. He uses his Shootout ability to become a Stud. He survives the shootout.
You use Good Stiff Drink on him. He may use that shootout ability again.
You experience him.
He no longer has his shootout ability.
He now has 2 noon abilities.
You may use both of those abilities for the 1st time (Good Stiff Drink has no impact on those abilities whatsoever), but you cannot use the shootout action as it is no longer in play.
Say you use both of those abilities, and play another Good Stiff Drink.
You'd have to choose which of the abilities you want to use again when you play Good Stiff Drink. It would not let you play both a 2nd time.
Nor could you choose the Shootout action as it is not in play to choose.