How to efficiently use time while preparing for a class?
I want to leave an answer that points in a different direction. You are a math graduate student (your profile says you are studying analytic number theory). Your teaching assignment provides you with an opportunity to improve your own understanding of a basic and important area of mathematics. I'd like to recommend that you take that opportunity and allow yourself the time you need to truly improve your mastery of the subject.
I am a 40ish tenured, full professor. I also work in number theory, which -- like many parts of modern mathematics -- is a very "cosmopolitan" field, drawing upon many different areas of mathematics. Over the years I have done research in various aspects of number theory. While I have some core tools and knowledge that have stayed with me over the years, there are a lot more things that I learned and knew better before than I do now, and there are still more things that I sort of learned as a student and would like to know better -- for my own edification but even directly for my research.
By far the best opportunity I have to learn something is in a course I'm teaching, at either the undergraduate or the graduate level. In teaching a course I am committing myself to several prep sessions a week for 15 weeks, and there are consequences involved in my slacking on this (i.e., the students!). There is really nothing else like this. So I have taken the opportunity to dig in and learn this material many times over the years.
For instance, currently I am teaching a senior level complex analysis course for the first time. I haven't thought about contour integrals since I was an undergraduate (which would be to my detriment in many parts of analytic number theory, as I'm sure you can appreciate), about 20 years ago. So I am taking the opportunity by putting in a lot more than the minimum prep necessary to give solid lectures out of the textbook chosen. At the moment I am a little more than halfway through the course and I have about 50 pages of typed "lecture notes"...of which maybe 25-30% are not actually being covered in the lectures. Perhaps I spend 3-5 hours a week more than would be needed to get by.
Having said that, of course moderation is key. For most tasks that you do in academia you have to set firm limits on the amount of time you will spend, or each individual task threatens to take all your time. That will never do. In your case, you should ask "How much time do I want to spend digging into MVC in the context of this TA assignment? How much time can I afford to spend?" This is a good thing to talk about with your advisor, as it largely depends on what else you are trying to do.
One more piece of advice: no one said that the time you spend on various things needs to be the same from semester to semester or even from week to week. When it comes to teaching in particular: the first time you teach a class is somewhere from 2-5 times as much work as when you teach it each subsequent time. So one key question is: what are your teaching assignments going to be in the future? Will you be TAing MVC again? If so, then spending more time on it now is justified by the fact that you'll have a much easier teaching assignment later on. (Pro tip: if it's not clear whether you'll be teaching the same thing again, if you want to, ask for it.) When I was a PhD student I taught second semester calculus in the fall semester four times, and that was all but one of my teaching assignments.
When it comes to teaching/TAing a course, the real work comes when you have to learn unfamiliar material well enough to stay ahead of the students. If you wait until that happens, you're going to find yourself having to commit lots of hours on short notice. So I would recommend tackling it in advance. As a graduate student, you can learn MVC much more quickly than an undergrad: an intense month at the beginning of the semester could power you most of the way through. Most advisors will be very understanding if your teaching responsibilities are eating up your time for a few weeks. (To be really honest about it: who among us has not wanted to go easy on the research front for a few weeks at a time? Digging deep into your teaching can be a way of recharging your batteries. It worked for Richard Feynman...)
Anyway, good luck, and sounds fun. I haven't thought about curvature and torsion in years. Hmm...
Set a time limit for yourself. Decide what a "reasonable" amount of time is to review the materials, keeping in mind your other obligations and priorities. Get as much done as possible in that time. Accept that you may be asked something that you do not know and that you will have to tell the student you will get back to them because you don't have a great way to explain that off the top of your head.
I TA'd for that exact class last year and it was a nightmare, primarily since I'd taken that class at a school that wasn't as good. But that experience taught me a lot about efficiency. The next quarter, when I TA'd a different class that I also had taken at a different school, I was ruthless about only brushing up on the things I knew the students would ask me about: the homework and the tests. Instead of spending hours studying every corner of the material, I just read the answers to the homework and tests. It kind of feels like cheating, but you're not the one taking the class. The point is to do whatever helps the students the most.