Does Flurry of Blades work on Barron?

Yes — Flurry of Blades hits Baron Nashor, though it won't kill him on its own.

Flurry of Blades doesn't target or "choose" units; it programmatically affects all units at battlefields (Rule 352.10.d, FAQ #1583). Baron's protection only stops enemy spells/abilities from choosing him, so the 1 damage applies. Since Baron has 12 Might, he just takes 1 damage and survives unless the damage is lethal-modified (e.g., you control Elder Dragon).