Once Obzedat's ability has returned it to the battlefield, it has haste for as long as it remains on the battlefield. If an opponent gains control of it (perhaps with Olivia Voldaren or Soul Ransom), it can attack that turn.

If you gain control of Obzedat "until end of turn," you'll control it during your end step and you choose whether to exile it. If you do, it returns to the battlefield under its owner's control at the beginning of your next upkeep, not its owner's next upkeep.