Does Gutter Palace's "win the game" trigger go through if Gutter Palace was given temporary?

Yes — it still wins. Temporary and Gutter Palace's ability both trigger at the same moment (the start of your Beginning Phase), and Gutter Palace is still on the board when that condition is checked, so its win trigger goes on the chain. Killing it afterward via Temporary doesn't stop the win.

Per FAQ #11528, Gutter Palace's "exactly 4 cards in hand and 4 units at battlefields" clause is part of the trigger condition and is checked when the trigger is placed on the chain, not on resolution. Temporary is just a triggered ability (Rule 816) — it kills Gutter Palace when it resolves, which is after the win trigger is already finalized on the chain, and it doesn't change your hand size or unit count anyway. The win trigger then resolves and you win.

The only case where Temporary would matter is if the 4-cards/4-units condition were already false at the start of the phase — then Gutter Palace never triggers and you don't win regardless of Temporary.