When opponent enters showdown against my defending Fiora, Peerless, does her ability trigger before or after I cast Grim Resolve (I cast Grim Resolve when the opponent enters the battlefield)?

No — Fiora's trigger resolves before you can cast Grim Resolve.

When your opponent's unit enters the battlefield, the Combat Showdown opens and Fiora gains the Defender designation immediately (Rule 464.2.c.3). Her "when I defend one on one" trigger goes on the Combat Chain at that exact moment, before any player can play anything (Rule 464.2.e). That closes the State, and during a Closed State only Reaction cards are playable.

Grim Resolve is an [Action], so you can't play it until the chain empties and you have Focus in an Open Showdown — by which point her Might is already doubled to 6. You can still play Grim Resolve then; the +3 Shield just applies on top of her doubled Might.