If Dodonna is on Garel's Honor, Garel's Honor is one damage card away from death, and Garel's Honor rams an enemy ship...does Dodonna's ability resolve before Garel's Honor is destroyed?

Your damage card and the enemy damage card are applied at the same time sou you as the activating player chose the order, so yes I believe he can effect things.

Timing is everything, Dodonnas ability would still trigger Garels Honor upgrade card, as the card is triggered during the ram, before the ship is removed from play.

Applicable Rulebook Quote:

If a ship executes a maneuver and its final position would overlap another ship, it cannot finish its maneuver normally. Instead, temporarily reduce its speed by one (without changing the speed dial) and move the ship at the new speed. This process continues until the ship can finish its maneuver, even if that maneuver is to remain in place at speed “0.” Then deal one facedown damage card to the ship that moved and the closest ship that it overlapped.

The Italic section is key there. The damage cards are basically happening at the same time (as it is an and and not a then ), thus, as stated - you as the owner (initiator) of the effect, choose the order of their resolution... Hopefully, triggering Garels, then Dodonna, then your own Death.

Maxim #1 - Pillage. THEN Burn.