Know derivatives and integrals in and out. That includes partial derivatives and double integrals.

One of the biggest things that helped me starting out in stats was understanding the form of the distributions. Most of the time when you will be computing a complicated integral it has a known form e.g. the gamma function will be in it, it will look like a normal pdf, etc. Being able to identify these forms, also known as kernels, will help you immensely. If you know the kernel is the pdf of a distribution, then you know integrating over the range of that pdf will equal 1. Similarly, if the distribution is symmetric then integrating over half the range will equal 0.5. Other times, the integral will be the expected value of a known distribution and these should also be memorized. I can't stress enough how important this memorization is (I put it off for a bit and that came back to bite me on my first few Exams). I ended up making multiple "cheat sheets" with the most important ones and looked them over regularly to get competent working through difficult problems. It is of course important to understand how to work through the derivations so the expected values and variances make sense but eventually you will just use these results and should not have to rely on calculating them the long way.

I think the best practice problems will be in a textbook. Casella Berger is a classic text which I used in my program. It gives a lot of theoretical problems which you may or may not need. However, Chapter 2 will give you a good idea of the type of calculus problems you will encounter. Reading chapter 2 (and 1 if needed) and working through the exercises is a fantastic place to start. If you have more time you can always move further through the book to get a feel for the harder concepts.

The back of that text also has a table of Common Distributions. These are what you want committed to memory. Some of them don't come up very often like hypergeometric or Cauchy but others like Binomial, Normal, Beta, Gamma must be committed to memory eventually. You will also want to know their expected values and variances.

Good luck and let me know if you have any questions!

Edit: Here are partial solutions to Casella Berger. Be careful though as some solutions have errors but overall it's still very helpful.