The exiled card is played using the normal timing rules for an artifact, as well as any other applicable restrictions. For example, you can't play the card during your combat phase unless it has flash. Similarly, if the exiled card is an artifact land, you can't play it if you've already played a land that turn. If it's a nonland card, you'll have to pay its mana cost. The only thing that's different is you're playing it from the exile zone.

If you don't play the card before the turn ends, it simply remains exiled.

When the triggered ability resolves, you must search that player's library (and they must shuffle it) even if there are no artifact cards in it or you choose not to find one.