Once Armory Paladin’s last ability resolves, it doesn’t matter what happens to Armory Paladin after that. You’ll still be able to play the exiled card until the end of your next turn.

You pay all costs and follow all normal timing rules for a card played this way. For example, if the exiled card is a land card, you may play it only during your main phase while the stack is empty.

If the card exiled with Armory Paladin’s last ability is an Aura or Equipment card, casting it will cause Armory Paladin’s last ability to trigger again.