As a native speaker of English, do NOT rely on ear! To my surprise, my ear gives worse results than most non-native English test-takers, and the reason is that they learn the proper way.

Pick up Manhattan GMAT's Sentence correction guide, and CLOSELY study it, and follow up with practice questions they provide. This has helped me out, though to be honest I'm still at ~80% on sentence correction questions. I think it has to do with the fact that when I can't recognize the grammatical error, I end up relying on ear.

You will be shocked to see how many answers you get wrong because of your "ear." The test makers are well aware of this, and purposely design the exam to test you on your ability to capture the rules and essence of the English language.

Do you think I have enough time to go through that book and really grasp what it teaches though in the time I have?

it really depends on how good your "ear" is. I have friends that never took grammar school but they are just good writers that "know" when a sentence is right or wrong.

on the other hand, if your "ear" is mostly from oral English, then I'd say hit the books and get some solid grammar foundation.

Best suggestion is to do at least 3 sets of sentence correction tests (about 60-70 questions) and see how well you fair. If you can get 80% correct, go back and see what mistakes you make and it might just be a couple specific grammar rules that you don't know. But if you get killed, you konw you'll need more time.