Mark (with post-its, for example) those items in the citation manual you found most surprising, and that you think you'll most need to be reminded of during the competition. By reading and marking the manual, you'll (a) get a good sense of the rules; (b) understand the general logic behind the rules (not all the rules are explicable using a general logical principle, but some are); and (c) see enough citation examples that you might more easily notice when something departs from the citation rules. Pay particularly close attention to the rules related to (1) cases, (2) statutes and constitutions, (3) articles, (4) books, (5) short forms, and (6) citation signals.
2. Ask your law review what style manual it uses, and read that, too, marking the surprising items.
3. Read a good general writing manual, such as Strunk & White's The Elements of Style, at least once.
4. See whether past competitions are available. Read them, just to get a feel for what's going on. If some model answers are available, pay particularly close attention to them.
5. If the past competitions include practice editing and proofreading tests, do as many of the practice tests as you can; compare your results against the answer keys, if those are given. If there are no answer keys, compare your answers against those of some friends of yours who are also doing the practice competitions. (You can't work together with people on the actual competition, but there's no problem with cooperating on practice projects.)
6. Go over any comments that you've gotten on your past written work, such as the papers in your first year legal writing course. Most writers make the same mistakes repeatedly. Figure out what your weaknesses are, so you can avoid them while doing the write-on.
Your writing instructor will likely be happy to help you with this. Writing teachers like it when you come to them out of a sincere desire to improve your writing; and they often have specific advice that they'll be glad to pass along.
7. Plan ahead to make sure you have no other obligations during your write-on competition. If you'll be working, even part-time, see if you can take the week off, and make up the lost time before or after. If you have children, do what you can to get the other parent or someone else to spend more time with them during the competition.
Try to avoid leaving town to see friends or family, even if it is your one week of vacation before you start your summer job. You might intend to do lots of work when you're on the trip, but it's hard to work when you're around people you haven't seen in months, and who understandably want your company. Going out to dinner with friends is fine; everyone needs a study break. But try to avoid more demanding commitments.
The writing competition requires you to do something that's new to you, under considerable psychological pressure, in a limited time. You'll want to finish your draft as early as possible, so you can edit it as many times as you can. You really might need most of your waking hours to do this. Even if you've found that the first year of law school hasn't been as time-consuming as you were initially told, this week will be quite a burden.
If, however, you can't get out of your other obligations for the week, don't use that as an excuse to just sit out the competition. It's possible for you to do well even if you also have to travel, work, study, or mind the kids that week — it's just easier if you can focus solely on the competition.
book — which covers doing the write-on competitions, writing your student Note, writing seminar papers that you might do even if you don't make law review, and more — that I'll be happy to personalize for you or friends of yours.