if I move akali deadly weapon onto a battlefield with overzealous fan, does the fan die before the showdown begins?

No — the Fan can't die before the Showdown begins from its own ability.

Overzealous Fan's "When I defend" trigger only exists once it actually gains the Defender designation, and designations are only handed out when the Combat Showdown opens (Rule 464.2.c.3, 383.4.f.2). So the real order is: Akali's move completes, her "When I move" damage resolves first — 1, or 2 if she's Empowered — against a Fan that still has no defender bonus (the Showdown doesn't open until the next Cleanup after that chain empties, Rule 344.2). Only then does the Showdown open, the Fan become the Defender, its trigger go on the combat chain, and the Fan be killed to pay for that ability and move Akali to base — all before any combat damage.

One caveat: if Akali is Empowered, her 2 damage is already lethal on the 2-Might Fan, so the Fan dies in the Cleanup following her trigger — before the Showdown opens. In that case no Combat Showdown ever forms (no opposing units remain), you just get a Non-Combat Showdown, and the Fan's ability never triggers at all.