I have unit with Blighted Battleaxe that has not conquered this turn, but I had punch first on a unit +5 might. Will it die at the end of turn?

No — it survives. Punch First's +5 Might is still active when Blighted Battleaxe's end-of-turn trigger resolves, so 4 damage isn't lethal.

Blighted Battleaxe's trigger fires in your Ending Step, which happens before the Expiration Step where "this turn" effects expire (Rule 317.2.c). So at the moment the axe unattaches its +4 and deals 4 to the unit, the unit is at 9 Might (4 printed + 5) with only 4 damage marked — not lethal, since lethal requires damage equal to or exceeding current Might (Rule 143.2.a).

Note the timing wrinkle: Punch First is an [Action] spell, so it must have been cast earlier in the turn during an Open State (e.g., a showdown) — you can't cast it in response to the axe trigger, since that chain is Closed and only Reactions are legal. If the Punch First was from a previous turn, the +5 has already expired and the unit dies. Also, the damage gets healed in the same turn's Ending cleanup regardless.