If the spell you cast has  in its mana cost, you must choose 0 as the value of X.

You cast the copy while the ability is resolving and still on the stack. You can't wait to cast it later in the turn.

Once a prowess ability triggers, causing the creature to lose prowess by removing Narset, Enlightened Exile won't affect that ability. The creature will still get +1/+1 until end of turn.

If you cast a spell "without paying its mana cost," you can't pay any alternative costs. You can, however, pay additional costs, such as kicker costs. If the card has any mandatory additional costs, you must pay those.

If a creature has multiple instances of prowess, each triggers separately.

Narset's first ability gives it prowess while it's on the battlefield.

If another effect causes Narset's power to be less than or equal to the mana value of the target card as the ability tries to resolve, the target is illegal. The card won't be exiled, and you won't get to cast a copy.

If you don't want to cast the copy, you can choose not to; the copy ceases to exist the next time state-based actions are performed.