You cast the exiled card while the ability is resolving and still on the stack. You can’t wait to cast it later in the turn. Timing restrictions based on the card’s type are ignored.

Since you are using an alternative cost to cast the spell, you can’t pay any other 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 the spell you cast has {X} in its mana cost, you must choose 0 as the value of X.

If you choose not to cast the exiled card, it remains in exile.