Are older copies of current standard-legal reprints playable again?
It doesn’t matter if a card is a reprint or being printed for the first time. A simple and obvious example would be a basic land card; a Mountain would be a reprint of an old card; but it would be standard legal because it was printed in a recent set. 

However, cards printed in Duel Decks are not necessarily standard legal. Duel Decks contain some cards that aren’t part of the current or recent block. Standard currently (in January 2018) consists of cards printed in Ixalan, Amonkhet, or Kaladesh block. 

To determine if a specific card is legal, you can check Gatherer to see the listing.