does Akali Deadly weapon ability damage happen before a unit can get +1 from defending

No — Akali's "When I move" damage lands before the defending unit ever gets its +1 defending bonus.

The "+1 from defending" is the Shield keyword: "While I am a defender, I have +X [M]" (Rule 814.1.c), and being a defender means holding the Defender designation, which is only granted when the Combat Showdown opens (Rule 464.2.c.3). But the Showdown doesn't open until the next Cleanup after the Chain empties (Rule 344.2). Akali's move trigger resolves first, on the Chain, while the enemy unit still has its normal Might — so a 3-Might defender takes Akali's 1 (or 2 if empowered) damage at 3 Might, not 4.

If that damage isn't lethal, the Showdown then opens, the unit gains Defender, and the +1 kicks in for combat damage and the Might comparison — but the Akali trigger damage itself is already marked. Rule ref: 814.1.d, 464.2.c.3.