My opponent moves Ambessa Respected and Feared into my Nocturne, they choose it to kill and I flip Switcheroo in response, does the kill still go through?

No — the kill doesn't go through.

Your hidden Switcheroo is playable here (Hidden cards gain Reaction when flipped, so you can respond during the Closed State, Rule 811.6), and it resolves before Ambessa's trigger. Switcheroo swaps the Might values, so when Ambessa's attack trigger resolves, "kill an enemy unit here with less Might than me" re-checks its target — Nocturne no longer has less Might than Ambessa, so it's no longer a legal target and the kill instruction is ignored (Rule 359.3.e.5; FAQ #7925).

One thing to keep in mind: if Ambessa is attacking with her Assault 2 active, Switcheroo still swaps the current totals (7 ↔ 4), so the same result holds — Nocturne ends up with the higher Might and survives.