Here's my comments on how to do well on the writing competition (I am on my school's law review, fwiw, but have no other credibility on the topic--just my opinion here):
1) FOLLOW THE RULES. seems like common sense, but be sure you hit the page limits, put the page numbers in the right place, use the correct font, etc. follow the rules down to the letter. this is typically one of the columns on the grading form.
2) read all of the materials before beginning to write. this may seem like a lot of work, but many people selectively read sources and may miss key pieces to the argument in hopes of getting done faster.
3) outline your essay: two or three times, using different arguments each time. leave them alone for a day, then reread and pick the best one to write out.
4) write simply. don't use big words simply because you can. use short, simple sentences when possible. be grammatically flawless. try reading your paper out loud to see if any of your sentences are so long that you have to take a breath in the middle of reading them. if you do, try to cut them in half (it's a good indicator that your reader will lose track of the point of the sentence before he's done reading the end of it). finally, use a book like "Legal Writing by Design" to pinpoint verbose phrases such as "whether or not" -> whether, "the fact that"->that, etc.
5) give yourself enough time to edit at least twice and proofread the hard copy at least once before submitting (just these three edits will probably take you a full day).
don't stress out too much over it. yes, it's a big deal, but life will go on either way. law review is a lot of work, so if you burn out and don't finish the competition (or aren't selected), don't feel like you're missing too much
edit: one thought on the bluebook exam, while I'm at it. If you have a week between the end of finals and beginning of the competition, read the main rule for each section of the bluebook and tab the following sections: 1) typefaces for law reviews, 2) order of authorities, 3) case abbreviations (appendix), 4) abbreviations of periodicals. Understand the rule of 5.
And the real key: look up every single citation if time permits. never "eyeball" it. Our bluebook exam only tested the rules which, while tricky, could be found in the bluebook (once you're on LR, you'll be disappointed to learn that there are still lots of areas where the bluebook hasn't yet created a rule--they probably won't put these sorts of issues on the exam). Also, check the actual text for errors as well as the footnotes (the bluebook covers capitalization (they love to incorrectly capitalize the C in court), ellipses, etc).
I think I'm done now