If my legend is Master Yi, I have a Fiora Peerless Duelist alone on a battlefield, and my opponent starts a showdown with only one unit, can I get the +2 might from Master Yi first before Fiora's might is doubled? What happens if I play a Punch first after the showdown starts? Would Fiora double that might as well?

Yes — Fiora ends at 10 Might, and no, Punch First's +5 doesn't get doubled.

Your Master Yi (Wuju Bladesman, Starter) gives a passive +2 Might the moment Fiora is designated as the lone defender, before anything hits the chain (Arithmetic layer). Fiora goes 3 → 5, then her "when I defend one on one" trigger resolves and doubles her current Might: 5 × 2 = 10, not 8.

For Punch First: it's an [Action], so you cannot play it while Fiora's trigger is on the chain — the game is Closed and only [Reaction] cards are legal (FAQ #8421). You have to wait until the chain empties, by which point her trigger has already resolved on her 5 Might. Playing Punch First then gives her +5 for a final 15 Might; the doubling does not re-apply to bonuses added afterward (Rule 432.1 — doubling uses the value at the moment it resolves).