If I played Premonition during a showdown and I use 3 gold tokens for the power cost, would my Prize of Progress get +3?

Yes — Prize of Progress gets +3 Might.

Each Gold token is a separate Gear token, and each activation (Kill this, [T]: [Reaction] — [Add] [A]) is its own "use of an activated ability of a gear." Using 3 Gold tokens to pay Premonition's 3 Power means you activated gear abilities 3 times, so Prize of Progress triggers 3 times for +1 each (Rule 357.1.a allows using Reaction/Add gear abilities during the Pay Costs step; triggered abilities can enter the chain in a Closed State per Rule 383.3.c). The +3 lasts until end of turn regardless of when during the showdown it happens.