Although they call it a grammar section, it's actually more of a vocabulary test. There are a few questions where they test your knowledge of passive, causative forms and polite language, but they are not a major part of the test and you can get away without a superficial knowledge of them. The best way to learn grammar/vocabulary is by learning short phrases rather than individual words. I had a terrible time remembering the difference between koto and mono until I memorised a few sentences that expressed the meanings of these confusing words.
There is a list of all the grammatical points which have appeared more than three times on the previous tests. The most important thing is time distribution. I like to do the short reading passages first because they are easier, and I can build up my confidence. Read the questions first. Especially the final question, which typically requires you to summarise the article. Just having seen the vocabulary and having been able to imagine what the story might be about from the words, gives you a big hint about the passage's meaning and gives you a sort of foothold for understanding the passage. You'll be surprised what a difference it makes.
Another important strategy is to familiarise yourself with the sort of reading passages and questions that are going to be asked. Common themes for the reading passages are: how the author learned something about himself or a friend or family member; a scientific explanation of something; a letter to a friend (They usually ask why the letter was written); a question about a graph (These are easier than they look); a question where you have to put a scrambled reading passage in order.
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