I joined McKinsey as an undergrad summer intern with an engineering background and so I know how you're feeling. Eventually I became a final round interviewer for McKinsey summer interns and so hopefully I can give you a balanced perspective. Frankly, top consultancies are not looking for people with a business background/prior knowledge and rather strong problem solving intrinsics together the various FIT dimensions. Specific to cases, I would say the following:
+ Intellectual curiosity: start immersing yourself with different types of businesses/companies and get in a habit of thinking deeply about the challenges they might be facing. I personally got an Economist paper subscription which is weekly and takes away the pressure of having to constantly read the news and something I would flip through when I had the time each week. Related to the second point but try building a structure around the key issues and developing an approach around how you could solve the problem and the likely recommendation. As a consultant, you should always have a perspective, distinguishing the know vs. unknown, where it's never too early to start!
+ Problem solving intrinsics: a common oversight is to spend the majority of your preparation memorising different frameworks (familiarise yourself but problem solving is about developing a framework from first principles that is relevant) over actually understanding WHY consultancies test for structuring, synthesising, quantitative analysis, conceptual brainstorming, etc. There's a lot of material out there but Victor Cheng's book is a comprehensive summary which I highly recommend as opposed to picking up bits and pieces from different people. An experience McKisney interviewer will also be able to give you a comprehensive download in an hour which might be a more efficient way of prioritising your time!
+ Effective practice: I was very fortunate to receive case coaching and be mentored by the McKinsey consultants I met on campus during the recruiting process. Finding a good balance of self prep, case partner practice AND case coaching with an experienced consultant is super important. I have come across many candidates who have done loads of case preparation but unfortunately incorrectly where they end up shooting themselves in the foot. Let me reiterate the importance of understand the WHY as well as getting coaching from someone who has been successful through the selection process at that consulting firm and/or an experienced interviewer who has been part of the selection process itself.