Can I play an Action card at the end of my opponents turn?

Yes. When an opponent is about to draw their card for the end of the turn, say: "Before you draw your card I am going to play an Action called __________" and perform the Action. A player must always give their opponent a chance to play an action card before the end of the turn, this is called Priority. If a player draws a card before their opponent gets the chance to declare their action card then the player must return the drawn card back on the top of their deck (or back into the shop) and allow for the other
player to resolve their Action card.