Strangely enough, core work really helps with handling hills. No need to do the fetish amounts that some folks do, but 20-30min of core work per week (maybe spread over 3-4 different days) helps.
When going uphill, don't look at the top of the hill or down at your feet. Keep your head aligned with your torso, as you do on the flat--otherwise your posture will go, and you'll dissipate energy that could be moving you forward.
As with all running (flat or hills), you need to maintain relaxation in the elbow. As it swings behind the torso, the arm should open a little bit at the elbow; then it will close by the same amount as it moves in front. This is a perfectly natural movement that should occur at any speed (though the opening/closing will be very slight at slow speeds). It's not anything you have to *make* happen; just keep your elbows relaxed and *let* it happen.
In line with this: everything that moves you forward happens under and (mostly) behind you. If you're "using the arms" to move up a hill, focus on the *backward* swing of the elbow, which should move well behind your torso. If you're pumping your arms hard, but they're basically in front of your torso the whole time ('cause we like to see what our hands are doing!), they're actually making the hill *harder*. Who needs that?
[In any case, your arms themselves shouldn't get tired, because it's the muscles of your torso that actually move them! If your arms themselves are getting fatigued, it's likely because you're tensing them unnecessarily. Maintaining relaxation in the elbows helps prevent that. Funny: tell runners to relax the arms, and often they let the hands flap at the wrist, which locks the elbow up more! Your wrists can be fairly firm, but the elbow has to have a bit of play.]
Anyway, some guys (often littler guys) are just better at hills. I wasn't great on uphills, but still beat a lot of guys on hilly courses--because I noticed that a ton of them would really work when they went up the hill, then take a bit of a breather at the top, where I'd go by them. Charging up hills may be good in training, but it's dumb racing. You want to just keep your rhythm up the hill and be ready to *speed up* when you crest it. After all, when you hit the top it's flat there and you *should* run faster, right?
I know, that sounds like a joke because at the top of a tough hill you feel terrible for a few seconds. But at the top of that hill you're going to feel crappy no matter what you do: run, jog, walk, or stand, you're going to feel bad. So why not race when you're at the top? If you're going to feel bad regardless, why not get something accomplished?
Remember that you're trying to beat the other guys over the course of the whole race, not just the uphills. Just try to "maintain" on uphills--it can almost feel like the other guys are towing you up the slope--but when you hit the top, *shift gears* and go by them. An easy way to shift gears: as you're cresting the hill, switch your brain from your legs to your arms. Pick one arm (doesn't matter which) and just count ten backward beats of its elbow. That corresponds to twenty actual strides--by the time you've done those, you should be back to running normally.
Sorry this is so long-winded--don't have time to make it shorter! Good luck. Keep us posted on your progress, please.