Does Red Akali's movement trigger deal damage before Shield becomes active?

Yes, Akali's movement trigger deals damage before Shield becomes active.

Here's why: When Akali, Deadly Weapon moves into an occupied battlefield, her "When I move" trigger fires immediately and resolves completely (dealing 1, or 2 if Empowered). Only after that chain empties and a Cleanup runs does the Showdown officially begin — that's when units gain the Attacker/Defender designations (FAQ #11684). Shield is a passive ability that reads "While I am a defender, I have +X [M]" (Rule 814.1.c), so it only applies once the unit gains the Defender designation during combat.

Since the damage from Akali's move trigger lands before the Showdown starts, the defending unit hasn't become a defender yet and Shield is not active — the damage hits their base Might. If that damage kills the unit, they won't be around to defend at all.