About one third of my students are not native English speakers. I would say that actually on the whole they are most comfortable with sentence correction. Yes sentence correction has rules but these are grammar rules that non-native speakers usually learn in their English courses. In fact, with the exception of idioms (which you mentioned), non-native English speakers often know the rules of grammar better than their English speaking counterparts who take these things for granted. Sentence correction does not have that many rules per say and it actually does not have that many areas of grammar that are tested.
The eight areas of grammar that are tested are found in the Veritas Prep VAMPIRES acronym. Of those categories, I particularly suggest that you stress these five common errors that are tested over and over again.
1. Number Agreement (including subject-verb, pronoun antecedent, etc.)
2. Misplaced Modifiers
3. Verb tense
4. Parallel elements/ Comparisons
5. Relative Clauses
Critical reasoning and reading comprehension can present more of a challenge for the non-native English speaker because just understanding the stimulus is not the goal as in sentence correction but rather the student attempts to answer questions about critical reasoning or specific portions of a reading comprehension passage. However, these sections are not as diverse as they seem and can be broken down into certain common question types.
You mention the quantitative section as being a strength. Many of my non-native English speaking students find that interpreting problem solving questions accurately can be a challenge as well, so do not take the quant. for granted.
Don't be too discouraged, you can succeed!