The real difficulty with books like Spivak is not that there is specific prerequisite knowledge a reader needs to have, but that they need to have good mathematical ability and be able to adapt to a different kind of math in which proofs and non-routine problems are really important. I'd say that if a person with just a normal precalculus background is of high ability, then they can directly use Spivak as their first rigorous math book. Contrary to an opinion I commonly see expressed, I don't believe prior knowledge of non-rigorous calculus is particularly helpful, and Spivak certainly doesn't assume it in his book.
But it's also true that many people will be able to use Spivak only after additional preparation. In ideal circumstances, I would recommend first reading a few books on other, more elementary topics, such as combinatorics ("Mathematics of Choice" by Niven), elementary number theory ("Invitation to Number Theory" by Ore) and inequalities ("Introduction to Inequalities" by Beckenbach and Bellman), as well as other subjects. This gets you used to thinking mathematically, and since these are interesting topics in their own right, the time is by no means wasted.
But since you're already in university and have good reason to be more interested in calculus, you might want to move faster rather than slower, perhaps just reading the book on inequalities as preparation.
A good test of whether you're already ready now for rigorous calculus at the level of Spivak would be to try to work through the introduction to "Calculus, Vol. 1" by Apostol, including the problems. If you're reasonably successful doing that, then you are probably ready for Spivak (or the rest of Apostol).