I'm commonly landing 20-24" browns where I go fishing. How would the 390 work on fish that size?
Just fine. Head over to Tenkara-Fishing for a picture from Rob Worthing of the Tenkara Guides of Salt Lake. He has pushed a 390 beyond all level of reason.
I will add though that fish of that size will take practice and a steep learning curve for most to A: land the fish and B: not break the rod.
The rod can do it but it does take finess to know the point inbetween carbon fiber explosion and fish running away with your fly.
The reel-less (and therefore drag-less) system is part of the challenge to me. I commonly land big fish on 4lb line, but I have the ability to play the fish a lot more than Tenkara allows.
Breaking the rod landing a fish is my worry
Not near as thrilling as fiberglass. but still fun. Rob actually called me before that trip to see what I thought the rod would handle. He was chasing 30" and up fish that trip.
I think you will be surprised at the size fish you can bring right in with far less fight than with a reel. You can put a great deal more pressure on the fish and end the fight before they are totally exhausted. I regularly catch fish in the 20" range and the longest fight I have ever had with Tenkara lasted less than 90 seconds.