Already a lot of people have substantiated that computers are not needed to learn computer science.
I would like to clear another misconception that you possibly might be having. Computer Science is not the Science of Computers. It is the Science of Computation. It is about solving problems using structured thinking and applied logic.
So don’t be disheartened. Learn the course, solve every problem you come across, critically analyse the algorithms, understand why there are so many different approaches to solve the same problem (Sorting is the classic example). Somewhere along your journey you will definitely need a computer of you own (not just university computer). But by then you wouldn’t even have the need to ask this question.