You pay all costs and follow all normal timing rules for cards played with Unstable Amulet's last ability. 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 Unstable Amulet leaves the battlefield before you play the most recently exiled card, you can play that card for as long as it remains exiled.

Unstable Amulet's second ability counts any spells cast from zones other than your hand. These are usually spells cast from exile, the graveyard, or the command zone. It also counts spells cast from outside the game, such as spells cast with Wish or Garth One-Eye's ability.