I do not believe there are any good such strategies, especially if you are in a Ph.D. program. I think your question is an important one because the comprehensive exam system at some places conveys some need to desperately learn lots of mathematics quickly, and as a result, many students feel this sort of pressure. The late Ken Appel told me once that nobody ever failed a comp on a hard question, but the grad students were almost always, in my opinion, scrambling to learn too much too quickly and then would fail on these easy questions because they were not able to slow down and adequately solidify the basics. In the end, I think most of the failing grad students were even more demoralized in the end because they felt that they "didn't even know" the basics when of course they could not when they were trying to learn everything all at once. I think there are a lot of grad students at a lot of places that really should not be in grad school, but universities have graduate programs and courses that need TAs and so many very weak students are let into grad school and spend a lot of valuable time working toward a degree in order to find themselves in a desperate career conundrum later on.
The reality is that students with solid backgrounds are building on a stock of intuitions and habits that have been built over years of practice. Your question is like asking how to quickly become fluent in a new foreign language. There are ways to improve quickly, but I don't think anybody really thinks that any such method will, for most people, quickly build fluency.
There is another strategy in this situation, though. Often when talking with some strong graduate students and mathematicians, you'll find that they disavow any proficiency with things, not in their areas of expertise, and deny having any ability to understand these things. Often these people do have the raw ability to learn more broadly, but instead seem to more completely embrace the understanding that their ability and time are limited and employ a "lens" approach to learning. They talk to people to find out quickly the way to think about the core fundamentals of a new area, and then develop this in a very targeted way. They master the basics unapologetically and stick close to what they know...gradually broadening the base of their knowledge.
The trouble with this, sometimes, is that you can finish a Ph.D. in this way, but if your results are not spectacular, you will face very stiff competition in the academic job market from people with solid deep results and a broader knowledge base. So it's a long shot when you have a weak background, and you have to think very carefully about whether you are in love with the idea of being a mathematician, or if you love mathematics. The first situation happens a lot more than you'd think, and it is unsustainable.
Another important thing to consider here is your health. If I were to tell you that you need to train to run a marathon in two weeks and you were by no means in shape for it, you'd probably not try to do it. Trying to push yourself to "survive" graduate school by trying to accumulate a bunch of things you aren't familiar with very quickly is similar, and just as trying to cram for a marathon would be bad for your health, doing something like this is likely to be very bad for your physical and mental health.
It is hypothesized that people are generally happiest and most successful when their skills match their challenges. I don't think grad programs are one size fits all, and the fit is very important to consider carefully.
A sprinkle of hope, though. Suppose you start with a good fit for a master's degree, and then focus on the basics, the rate at which you learn is probably roughly proportional to what you've already solidly learned and so exponential growth may make you more competitive than you originally ever thought you could be in a strong Ph.D. program. This probably works best, though, when you aren't thinking about volume or speed but are feeling safe enough to focus on what is in front of you.
