If I move red Akali to a battlefield where my enemy has a 1 might unit, my ability does 1 damage to his unit when I move to this battlefield, does my opponent can play a card action/reaction, or does it instantly die?

They can play Reactions, but not Actions — and if they don't react, the 1-Might unit dies before any Action window opens.

When Akali, Deadly Weapon moves, her "When I move" trigger goes on the Chain immediately, putting the game in a Closed State (Rule 354.1). In that state your opponent may only play cards/abilities with Reaction — Action spells can't be played because no Showdown is staged yet. So they can save the unit with a Reaction (moving it to base, preventing the damage, buffing its Might, etc.), but if they pass, her trigger resolves dealing 1 damage and kills the 1-Might unit on the spot.

Only after that chain empties and a Cleanup runs does the Showdown officially begin (FAQ #11684, #12081). Your opponent's first chance to play an Action comes inside the Showdown once it opens — and since you applied Contested, you get Focus first (Rule 345). By then the damage has already resolved, so without a Reaction the unit is gone.