Can I move irresistable faefolk targeting akali deadly weapon, then ambush in a unit in response to akali deadly weapon's damage before my faefolk dies?

Yes — you get that window. Akali's "When I move" trigger is placed on the Chain after Faefolk's own move-trigger resolves, and the damage isn't dealt until that trigger resolves. Nothing is marked on Faefolk yet, so you control a unit at that battlefield and can play an Ambush unit there as a Reaction (Rules 822.1.b, 383.2.c).

Two things to keep in mind. First, Akali's target is locked in at finalization before you ambush, so if she chose the Faefolk that damage still lands on resolution — your ambush doesn't redirect it. Second, the ambush must be a genuine Ambush (or otherwise grant Reaction/permission for that location); without it a plain Reaction unit can't be played into a contested battlefield (Rule 813.3.a).

And to be clear about the timing you're asking for: the Faefolk does die. It only dies in the Cleanup after the Chain empties, once the 1–2 damage is actually marked (it's a 1-Might unit, so either value is lethal). So yes, the ambush resolves and enters before Faefolk dies — the Faefolk just won't survive the cleanup.