Does using Akali legend ability have to be in a showdown only?

No, it doesn't have to be used only during a showdown. The Akali legend ability has the [Action] keyword, which is permissive — it allows activation during showdowns but does not restrict you to them (Rule 806.2). The default timing for activated abilities (your Main Phase, Open State) is still available.

However, practically you can only use it during a showdown. The ability targets "a friendly unit in a showdown" — since units are only "in a showdown" while a Showdown State is active, there won't be a legal target outside of a showdown. So you can't activate it outside a showdown simply because no valid target exists.

In short: Timing-wise it's not showdown-only (Action is inclusive, not exclusive), but target-wise it is — the ability only works when there's a friendly unit in a showdown to move.