Can i win a game if i have Gutter Palace in base. a sprite on board 5 cards on hand and a sprite at a battle field and when the temporary from the sprite triggers i play Flurry of feathers as a reaction to the sprites death) leaving me with 4 cards in hand and 4 bids on board. and then triggers the Gutter Palace ?

No. Gutter Palace's win condition is checked at the exact moment your Beginning Phase starts, when you still had 5 cards in hand — so its trigger never fires, and reacting with Flurry of Feathers afterwards can't create it retroactively.

Per FAQ #11528, Gutter Palace's "if you have exactly 4 cards in hand and exactly 4 units at battlefields" is checked when the trigger would be placed on the chain at the start of your Beginning Phase, not when reactions later resolve. The Sprite's Temporary kill and Gutter Palace's ability both trigger simultaneously at that moment; since the 4-card condition was already false, Gutter Palace never goes on the chain, and the Temporary trigger only gives you a Reaction window after that check has passed (Rule 383.3).

One extra note: even if the timing worked, the four Bird tokens from Flurry of Feathers must be played to your base or a battlefield you control — tokens at your base wouldn't count toward "4 units at battlefields." To win this way you need exactly 4 cards in hand and exactly 4 qualifying units at battlefields already present at the start of the Beginning Phase.