Athleticism by far. Look where all that shooting got Redick. Nowhere. Look where all that athleticism(and zero skills) got Tyrus Thomas. A top 4 pick in the draft, and a chance to go down as one of the five greatest Bulls in franchise history.
I think size is probably #1. Although you have some rare athletes like Wade and Iverson who are a little undersized, you usually see the taller guys go higher. The saying that you can't teach height is true.
I think height goes hand and hand with how athletic a player is. A guy can be a 6-10, 150 PF coming out after his junior year, but if he can't run and jump, he isn't going that high.
Of course you have exceptions to both, a tall, athletic player will go before a shorter, less athletic player 9 times out 10, regardless of talent. You can always teach shooting, passing, and defense.
I think shooting really doesn't come into play that much. Sure it is great if a guy has good form already, but you can teach a guy to become a better shooter (look at Michael Redd). I would put shooting about where I would put other skills like passing, mid-range and close shots, defense, rebounding, blocking shots, and stuff like that.
The hardest thing about making a draft pick, though, is knowing how much heart and work ethic a guy has. You can see if he can run, how tall he is, and how well he shoots, but it is harder to tell who will work hard enough to make them the best they can be. Some guys like Matt Harpring and Shane Battier you could see this in, but 90% of the other guys it is so hard to tell who will be the worker and play his heart out every game, and who will be the guy that just collects a check and puts on a uniform.
But to answer the question, I think you have to look at athleticism before shooting when only looking at the two head-to-head.
It's hard to argue which of these traits make a player better. Coming into the league, the rookie needs to be strong mentally and physically. They have to be able to understand the game, play their roles, and be able to take criticism. They should have good character and hang with the right crowd, or they will likely end up as journeymen. Then comes the player traits. Traits vary in importance from position to position. If you're a center, shooting is probably the thing you are least concentrating on. At guard, it is most likely the first in traditional offenses. Athleticism's importance has grown over the years, and is probably the most outstanding trait of a small forward. Height is important for pretty much every position. Most of the top point guards in the league are at least 6'0", shooting guards 6'4, small forwards 6'6", power forwards 6'8", and centers 6'10". Anything less than those measurements will probably not get a player on a team, unless they excel in other areas of their game. For a draft pick, the most important trait is probably ball handling for a point guard, shooting for a shooting guard, athleticism for a small forward, athleticism and strength for a power forward, and height/athleticism for a center. If I had to pick the most important trait for any position physically, I would probably have to say athleticism, since the NBA has started to shift to a more up-tempo game, and we are seeing power forwards go from being bangers to athletes like Chris Bosh. Height is also important, and I would probably consider it 1b.
Athleticism/Size gets the nod over shooting - scouts see a guy like Tyrus Thomas and immediately think 'damn, if this kid can learn to shoot a half-decent jumper he'll be unstoppable'. What it comes down to is that you can develop a better shot but you can't learn athleticism - you don't hear things like, 'if JJ Redick just learns to be taller, faster, stronger, how to block shots and how to jump higher, he'll be great'.
Redick is also the perfect example of a great shooter that isn't a great shooter in the NBA so far. He can't create his shot which is based on athletic ability/height, and also ballhandling. Just look at college basketball. There are tons of guys draining 3's at 40% clip, many of them deeper than the NBA line, but they don't do anything or even make the NBA.