Nightveil Specter - Do I have to play the cards in order they were exiled?
See the rulings for this card on Gatherer: Gatherer: Nightveil Specter


  
  The card is exiled face up. All players may look at it.
  Playing a card exiled with Nightveil Specter follows all the normal rules for playing that card. You must pay its costs, and you must
  follow all timing restrictions, for example.
  Nightveil Specter’s last ability applies to cards exiled with that specific Nightveil Specter, not any other creature named Nightveil Specter. You should keep cards exiled by different Nightveil Specters separate.
  




The cards are effectively put to one side, to be used as you see fit. It's as if they are added to your hand. You can play them when you choose, but you have to follow the usual rules for playing those cards, including costs and timings. So you're still restricted to playing one land per turn.

The order of exile isn't relevant here, only the copy of the Specter that exiled the card. So if it was exiled last turn it’s still available to you. This is because exile is effectively a place, the card has been removed from the game, and it stays that way until something changes its state… like you choosing to play it.