If I have a sprite that dies at the start of my turn due to Temporary, does Zhonya's Hourglass proc and save that sprite? If so, will the sprite still have the keyword Temporary?

Yes, Zhonya's Hourglass will proc and save the sprite. The replacement effect intercedes when Temporary triggers, killing Zhonya's Hourglass instead and recalling the sprite to base (healing and exhausting it). The sprite survives because base is a Board location, not a Non-Board Zone. The sprite retains Temporary because it never left the Board. However, this only buys one turn—next turn, Temporary will kill the sprite again since Zhonya's Hourglass is already dead.