I find that often, because of time pressure, students try to go quickly through CR by speeding through the argument, and then deliberating on the answer choices. As you say, this can often lead to misreading (or just missing) key pieces of information. Really, the best way to tackle CR quickly and effectively is to read the argument very carefully, and take a moment to think about what you're looking for. If you have a clear idea of what you're looking for in the answer choices, you can actually go through them much more quickly.

You'll want to know each of the major question types, and what to be looking for with each (you can search of the forum or guide for more information about each:
- Assumption - look for a logical gap between the premises given and the conclusion
- Evaluate - also look for a logical gap, and see which answer asks a question about it
- Strengthen/Weaken - again, find the logical gap, and find the answer that either upholds it or undermines it
- Draw Conclusion/Inference - think about what MUST be true given the premises
- Discrepancy - make sure you understand what the discrepancy is, and why it's a discrepancy. Look for the answer choice that keeps it - and the original condition - true.
- Argument Structure - make sure you know how to identify the main conclusion, supporting premises, and counterpoints

While you're working through the answer choices, pay attention to the exact wording of each answer choice. Focus on trying to prove 4 of them wrong, not pick 1 right one. Good luck!