I would consider the following:
More enclosed: If your developmental timeline permits it, copper for steep surfaces and slate for more level surfaces. A focus would be on steep rooftops so that dropped objects will fall to the ground rather than penetrate the structure. Coatings (e.g., rubberization or tar) might also be employed to protect surfaces from acids and other corrosives.
Protected Walkways: Where many of Earth's castles had an open space where all traffic, people and animal alike, traversed, I can imagine a focus on covered walkways, connected buildings, and even underground foot-traffic passages for everything but carts. (And maybe even some large ones for the carts.
Relocation: Earth's castles were often located because (a) the geography made them more defensible and/or (b) the location made observation of the surrounding countryside easier. In your case, I can imagine mountain sides such that only one side of the castle is easy to approach from the air. I can imagine taking advantage of natural caves. And I can imagine (b) not being an issue at all since flying troops would always be more efficient. Rather than locating near waterways, they would locate (perhaps) at altitude to minimize the number of attackers (or, said another way, to maximize the dependency on skill. Weak flyers can't easily reach the castle).
Central Towers: Where our castles tended to have narrow windows to maximize defense from arrows (and because glass was something that came later... large windows meant large drafts), I can imagine tower tops similar to helipads that permitted easy egress for stationed troops. Towers would be thick to accomodate the structure and to hold troops and supplies. Where our castles had (generally) some spires and high places for observation, your castle may have one or two cylindrical towers (basically a flat top, or some structure only on one side or dead center) about which the rest of the castle would be built.
Archers: Not knowing your technology level, I'm assuming that non-flying archers would find emplacement to protect against all but the highest-flying attackers. This might necessitate early development of smoked lenses (sunglasses) to protect archers against the sun as flying between sun and defenders would be an easy counter-attack.
Diminished Dependence on Walls: I'm guessing that having flying troops might cause less dependence on siege equipment. If so, sideways defense against incoming stones might give way to top defense against dropped items. Thus, your castles might not have a substantial surrounding wall (or it may be less prominent).
Avoidance & Detection: Finally, and I'm thinking about pidgeons here, your castles might develop roof spikes to keep unwanted guests from landing and weighted (if there isn't much snow) trip-pads where aerial attackers must land to enter a building that would activate an alarm (fancy way of saying, "ring a bell").
Searchlights: Finally, finally, I can easily imagine early development of basic mirrored optics. Think "searchlights" but with flares (brightly burning candles) vs. arc electricity. Of course, this would only be valuable if the searchlight can see the incoming attacker before the attacker is in range to destroy the searchlight... but it might be possible. Like I said... flares.