wich one resolves first: Blighted Battleaxe or punch first?

Blighted Battleaxe resolves first — and you can't even play Punch First in response.

Punch First is an [Action] spell, so it can only be played in an Open State. When the axe's end-of-turn trigger goes on the Chain, the state is Closed, and only Reaction cards can be played (Rule 334.1.a.1).

After the axe trigger resolves and the Chain empties, you're in the Ending Step — priority isn't granted outside the Main Phase (Rule 335), so there's no window to cast Punch First before the turn ends. And even if you could, the axe's 4 damage gets wiped by the turn-end "Heal all Units" cleanup (Rule 317.2.b).