How to teach a class that I've never taken?

It will be a huge timesaver if you can get lecture notes from someone who has taught the class before. There's a big difference between understanding the material well enough to solve problems and answer questions and understanding it well enough to find the clearest, most succinct way to present it. If you get someone else's lecture notes, you will only have to attain the first level of mastery, rather than the second; this will save you a lot of time.

Before the class starts, you should sit down with the syllabus (presumably you can get one from someone who's taught it before; typically, the first time I teach a class, I don't change much unless I have a good reason to) and try to understand the structure ("story arc") of the class. What are the big ideas? For example, in Calculus I, some of these are: limits, derivatives, integrals, applications. That gives you a sense of where you're going and helps you to know what may be important.

Don't feel like you have to know every little detail perfectly. The stuff that will likely stump you will be the obscure corner cases. The stuff that most of your students will struggle with will be much more basic, like the limit definition of a derivative, and precalc and algebra. If a student asks you a question that you can't answer right away, learn to be comfortable saying "that's a great question"; it's a little outside of the scope of this lecture, but I'd be happy to talk with you about it after class.

Thinking on your feet in front of an audience (especially about unfamiliar material) can be really tough. Most of the time, you can avoid it. Usually the student will be happy to talk with you after class. The reduced pressure of not being in front of the other students will help you to think more clearly. If you still can't answer the question after a reasonable amount of time, say "This is a good question. Let me think about it some more and get back to you in class next time." At that point, feel free to ask your colleagues. Often one of them will have encountered the question before, and will know the answer off the top of their head.

tl;dnr Get lots of help from colleagues who have taught the class before. Typically, it won't take much time from them, but it will save you a lot of time (later, when roles are reversed, be willing to do the same for someone else). You don't have to be perfect. If you have decent lecture notes, and you really engage with the students and answer their question, you'll be fine.

Dan C gave a tremendously good response to this question, but I thought I'd add one more idea: with the preponderance of online classes, it is very much possible to follow a set of lectures online as you teach a class for the first time (especially for introductory classes), and many times you can find full semester classes for free on YouTube, or Khan Academy, or MIT OpenCourseWare. I've done this for an Astronomy class that I taught, and I not only learned a lot from watching the class, but I was able to set up my own classes with a good idea of class-long topics. Furthermore, I've said it at least once on Academia.SE before: observing other teachers teach is one of the best professional development methods around; you learn the good and the bad, and you see what works and what doesn't work.

The reality is that sometimes, staying one step ahead of the students is the best you can do, even if it isn't ideal from a pedagogical perspective. As DR famously said, You "go to war with the army you have, not the army you might want.".

I know this answer might be controversial, but I think it is important to mention it. The solution is to not teach the class.

• Teaching is a huge responsibility: you are responsible for the enthusiasm (and possibly the career) of a lot of students. If you do not feel ready you are not ready.
• Good teaching takes a lot of preparation. You indicate that you are concerned about the time it will take you. My conclusion: you do not have the time to prepare properly. This is a good reason not to teach the class.
• You are looking for effective strategies. There is no such thing; teaching is not some kind of "trick". The next two points explain why.
• You need to have good (if not excellent) knowledge of the field: "If you can't explain it simply, you don't understand it well enough" can be reversed as well: if you do not understand the material well enough you will not be able to explain it simply. Reading a few course materials in the short time before the course will not be sufficient.
• Teaching takes planning. Planning takes time and experience (even if you are using existing course materials, you need to be very familiar with them and what they are meant to achieve).

Don't take the responsibility of teaching when you are not ready. But you can start preparing for next year, or ask to teach subjects you are more familiar with.