More than any other skill in ultimate—besides, maybe, getting layout blocks—dominating in the air has largely to do with raw athleticism, aggressiveness, and instinct. There's really no substitute for being a physical beast. If you're tall and fast, and you have springs for legs, you're going to posterize some poor souls, and in the process you'll develop confidence and intuition that you'll bring into every future matchup.
So: train. Even if you're not a premium genetic specimen, effective training can bring out the inner beast. You can add a few inches to your vertical and shave a couple tenths off your 40; perhaps more importantly, preparation gives you confidence. I find that if I haven't trained very well, then when a disc goes up, I notice doubts creeping into my mind. I second-guess how high I can jump; I wonder if my guy is faster or stronger than I am; I worry about being skied. But if I'm confident in my training and preparation, then when the long throw goes up, my first thought is, "It's mine!" And that's pretty much the only thought in my head until the disc is (or isn't) in my hand.
If you are going to be thinking while the disc is in the air, then one useful thing to concentrate on is your "launch pad"—the place you ideally want to be jumping from. The one-on-one contest for a long throw is actually two contests: the first is a race for position, the second a test of who can leap higher. To win the race for position, get your body between your opponent and the launch pad as quickly as possible. Then, ideally, you'll decelerate as you approach the launch pad, while using your body to box out/maintain position until the jump. (You want to put yourself into position to decelerate because you jump highest when you're neither sprinting all-out nor standing still). If you can win this contest for position, then it doesn't matter as much who can jump higher.
There are two good ways to prepare for this battle of positioning. First, when you're practicing long throws with a friend, try to catch each high pass at the highest point possible. This will help give you confidence in your own assessment of your leaping ability. In other words, you will practice figuring out where your launch pad is. (A common error in deep receiving/defending is that a player focuses on where and when the opponent is jumping, rather than the optimum point for his/her jump). Second, when you're playing games of "500" with a couple of friends of similar height, try playing with "no jumping allowed" rules. This will force you to focus on out-positioning, rather than simply out-jumping, your opponent.