What's your castle for? There are a few different reasonable answers:
1) Be imposing to look at and tower over the surrounding countryside.
2) Provide a secure strongpoint on a border.
3) Provide a refuge for people to flee to in the event of attack.
4) Somewhere for the rulers/rich to live.
(1) and (4) are the same as ever, except as (2) and (3) change things (although, if your ruling class happens to consist largely/entirely of reasonably strong flyers, their homes may be set up to be inaccessible except by reasonably strong flyers, to reduce how many people can get in).
Now, for the others, a lot has changed: for (3), the correct answer, where possible, is basically "just bury it". It doesn't matter how well your attackers can fly (except for the ones that are going ridiculously quickly, I guess) if they need to attack down a narrow tunnel. If this is your only use, you're ending up with a tunnel system with only very defensible (and probably mostly hidden) entrances. You might want some farms above-ground, and those are going to be very hard to defend: they've got to be open to the air, just by their very nature. 548
Honestly, I don't see a good way to defend this against a force with superior numbers in the air, unless you dedicate a large percentage of your population to just standing around there under some kind of protection shooting at those air forces. This might just be a matter of not having them, or giving up on them if the enemy has air superiority (therefore making air troops an essential element of any siege force that significantly shortens sieges).
For (2), it's still mostly a matter of "just bury it", but you still need to have something sticking up above the surface, as a watch tower if nothing else. So basically a tunnel complex as above, plus some towers sticking up that have no entrances and just some narrow slits to look out (too small for anybody to get in through).
If we're adding (1) in, pretty much as above, but make your sticky-up towers bigger and more imposing.
If we're adding (4), you might want a more significant above-ground complex (caves aren't that nice to live in). You still want it set up so that you can still get anywhere in the castle while staying underground except for right at the end, and certainly without going into the outside areas, and plenty of (well defended and easily sealed) entrances everywhere to let your rulers get underground in a hurry. 548