Can an ability be used after my opponent uses an instant?
Yes, you can do that. In Magic: the Gathering, most creature abilities are used like instants.

You should read the Basic Rules (PDF), page 10, if you're not familiar with the Stack and how spell casting works.

The situation you describe would work like this:


Opponent casts Victim of the Night, choosing your creature for the target. It goes on the stack and doesn't resolve yet.
Opponent chooses to cast nothing else yet and passes priority to you.
You use Bubbling Cauldron's ability. That too goes on the stack and doesn't resolve yet. You pay the cost immediately, including sacrificing the creature targeted by Victim of the Night.
You choose to pass priority so your opponent can respond. They choose to do nothing.


Now the top object on the stack resolves.


Bubbling Cauldron's ability resolves first. You gain 4 life.
Again there is an opportunity for your opponent or you to respond in turn, if neither of you do, the top object of the stack resolves.
Victim of the Night resolves. Its target is gone, so it can't do anything. A new target can't be chosen. Victim of the Night goes to the graveyard.