First, I'm a man, so I don't actually know these terms very well. I had to look them up, so correct me if any of them seem incorrect.
Doe eyes are eyes with deep sockets, more 'round' in the sense the middle of the top eyelid reaches up closer to the brow region and looks more 'open' because of it, and the bottom waterline is tilted downwards slightly/is more visible.
Hunter eyes are a bit more masculine in the sense the upper eyelid rests closer to the center of the eyeball and traditionally has more of the upper eyelid on display rather than creased/folded.
Fox eyes, to my understanding, are only ever really achieved with an eye brow lift.
If this seems to be correct, then in most cases, none of these factors alone will effect the cosmetic aspect of aging more than another by themselves. (Ignoring the fox eye.) Most of it would have to be how well you treat your eyelid skin throughout the years. How often you rub your eyes, if you use any eye cream there, if you move/shift your eyelid often like for eye contacts, and so on.
Due to the deeper crease/sockets of the doe eyes, this will probably be
less noticeable of the eyelid sagging (if you were to develop it), but I don't think it'd prevent it altogether. I think a big factor in it would be your body weight/body fat % and your eyebrow position.
You would get it fixed with a surgery called blepharoplasty or you could try double eyelid surgery if you're asian.


(blepharoplasty) - (double eyelid surgery)
If you wanted to prevent it, you would just stay hydrated, reduce sun exposure, keep moisture there if they have a tendency to dry out, general skin care practices, and/or do hyaluronic acid fillers/injections.