Does the bonus damage from this ancient wonder apply to auto-resolve outcomes? I mostly use those when I fight in provinces so was just curious.
Yes, in the same way anything else that improves troop stats are applied in fighting, automatic or otherwise.
On the AW side, that applies to Needles (LR), Heroes' Forge (HM), Dragon Abbey (Mages), Temple of the Toads (HR), and Victory Springs (LM).
Also important for fighters are the temporary military buildings such as Enlightened Light Range, Magnificent Mage Multiplier, Unleashed Unit Upgrade, Dwarven Armorer, and a smattering of various other sale or event-oriented buildings of entirely forgettable names. The first three can be obtained via crafting in the Magic Academy* while Dwarven Armorer can be obtained by getting lucky** in Spire.
* I personally craft them every single time they appear, even in cities where I don't even fight. Never know when you might need to switch from catering to nuking your enemies from orbit.
** If you play Spire often enough, "getting lucky" is only a matter of time when you're rerolling the dice as often as possible.