my unit has 4 might printed. This unit is equiped with blighted battlexaxe and did not conquer this turn. During the showdown i playd Punch First on it. Did my unit die for the 4 damage or de +5 might still conts?

No — your unit survives. The +5 Might from Punch First still counts, so the axe's 4 damage is not lethal.

Here's the sequence: Blighted Battleaxe's "at the end of your turn" trigger resolves during the Ending Step, before "this turn" effects expire. When it resolves, the axe unattaches (its +4 Might bonus stops applying, Rule 137.3.a) and deals 4 to the equipped unit — but Punch First's +5 is still active. Your unit is at 9 Might (4 printed + 5) with only 4 damage marked, which is not lethal (Rule 143.2.a). Without Punch First it would have died (4 Might, 4 damage); with it, it lives. (Damage also gets healed in your Ending cleanup anyway — FAQ #11352.)