If I have a creature that became monstrous, it gets destroyed, and I pull it from my graveyard, will I have to reapply monstrous or does it last until the end of the game?
A creature that leaves the battlefield stops being monstrous.
Whenever a creature (or any other object) changes zones for any reason, it becomes a new object. That means it will lose monstrous, among other things.
400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence.
Also, only permanents can be monstrous, and permanents only exist on the battlefield
701.31a “Monstrosity N” means “If this permanent isn’t monstrous, put N +1/+1 counters on it and it becomes monstrous.”
110.1. A permanent is a card or token on the battlefield. A permanent remains on the battlefield indefinitely. A card or token becomes a permanent as it enters the battlefield and it stops being a permanent as it’s moved to another zone by an effect or rule.