Ah, ARMY PT...quite the load of BS it really is. The APFT is no different. I heard they were looking at changing the standards here shortly. MTF on that part of course. As for improving your run, I do believe 400 meter interval sprints would be a good way to go, along with some tempo running. I've seen a lot of performance increases from doing timed interval work, I think some people call it tempo running. Pretty easy stuff, just set up a time frame to work on, say 30 seconds of sprints, 30 seconds of easy pace. Continue for desired distance. Then work your way up in time (ie. 60 seconds of sprinting: 60 seconds of easy pace).
For a really killer way to tax your body out try TABATA protocol, 20-30 seconds of sprinting with 10 seconds of rest. This one taxes me out everytime. Generally you can't do this one for very long before getting exhausted.
Hill sprints, how could I forget hill sprints. Find the biggest, baddest, steepest hill you can find that's about 50-100 meters and start trucking up that sucker. Want more, add a sandbag. That one's a vomit session for sure, but it'll get your conditioning up really, really fast. You can't excape the torture of hill sprints. Then we have sled drags, modified prowler pushes (I just use a car or small truck), sand bag shuttle runs, so many options. All of which will help you slam the PT test in the face.
Then as far as bodybuilding goes, well you're not going to get huge training that way, but you'll be lean. Actually, as long as you're eating enough and lifting fairly consistantly in the gym you should be fine, since most interval, high intesity training that I mentioned above doesn't make you lose a whole lot of lean mass. You may even be able to bulk up a bit. It the long distance stuff that will eat up your muscle tissue fast. Keep it under an hour with good supplementation / dietary protocol and you can pack on some good lean mass.
Personally, I like to lift before PT gets underway. I don't like the gym being crowded with too many people, and you can't get anything done during PT hours anyway. So I get up early, go lift, trying to schedule it around whatever sort of PT they have planned (ie, if you're going to do a 6-mile run I definately try to stay away from any serious leg training or lowerback training), bike home, slam a protein shake, maybe some carbs too if I'm drained and then go do PT. Seems to work pretty well. But it's also very individualistic as far as gains are concerned as with everything. You just have to experiement and figure out what works for you, whether it's five days of hard lifting and two days off or alternate days of lifting with PT inbetween those. Write it down in a log and figure out where your best gains are coming through. Stick with that until you stop making progress and re-evaluate. I know that won't help you in the short term, but one cannot figure ones body out in a day. Sorry, won't happen.
Hope you've found something useful in this book I just composed.