Can phasing a card out save me from paying an upkeep cost?
Yes. If a permanent with an upkeep cost phases out during your upkeep, you do not pay the upkeep, just as if it had been destroyed. For example, if you enchant your opponent’s phasing creature with Thirst, you will have to pay Thirst’s upkeep during your upkeep only if your opponent’s creature is in play–not if it’s phased out–because Thirst phases out along with the creature.