Spivak's Calculus and Calculus on Manifolds (for multivariable calculus) are pretty standard rigorous calculus texts. Rudin's Principle's of Mathematical Analysis is standard for a first analysis course, but may be too abstract for a physics course. If you don't like Apostol but still want the mathematical rigor, these are good alternatives.
There's also Strang's free calculus text here.
For more calculations, Hubbard's Vector Calculus book works as well.
Edit: Yes, it does depend on where you are from, since American students (such as myself) wouldn't study Spivak in our first year (to be fair, I actually never studied Spivak's Calculus, just his Calculus on Manifolds for a freshman advanced math course but this was far from the norm and only for math majors, not physics majors) and it is considered to be one of the best rigorous calculus textbooks as far as I'm aware. Seeing as the OP's professor mentions Apostol (although not mentioning which Apostol book he means, either his two Calculus books or his more advanced Mathematical Analysis), I think Spivak's Calculus books are a good alternative to Apostol's Calculus books, although they do not contain any differential equations theory or as extensive a treatment of linear algebra as Apostol (Calculus on Manifolds includes a bit of linear algebra in as far as it allows him to discuss total derivatives and multilinear forms).
All this said, I'd honestly need a bit more information on the course in order to give a proper alternative. There are far too many advanced "rigorous" vector calculus textbooks to choose from for such a general recommendation, so I gave what are considered by most people to be fairly standard references.
There are other books of course:
- Fitzpatrick's Advanced Calculus: Again, little in the way of
linear algebra or differential equations, but I think he explains
integration much better than Spivak and a bit more drawn out explanations, which I think is nice if you're new to the material.
- Boas's Mathematical Methods in
the Physical Sciences: This book probably has more material than you
need (PDEs, integral transforms, basic complex variables). However,
I'd say probably the first 400 pages is a great alternative to
Apostol, and as the title suggests it's good for a physics course.
- Stroyan's Foundations of Infinitesimal Calculus: Haven't read this one to be honest, but seems pretty good based on the table of contents, although it does seem a bit short.
- Protter's Basic Elements of Real Analysis: A nice springer undergraduate text which is an abstract version of Stewart's Calculus, but not quite as rigorous as others on this list. For more rigor but using a similar style, he also has a book called A First Course in Real Analysis, which I also like.
These are my best "general recommendations". Honestly, we do need me information to give good recommendations (does your class have a website/what is the class on/etc.).
Hope this helps.