Self-learning Calculus. Where does Lang's First Course in Calculus stay when compared to Apostol/Spivak/Courant

You ask an interesting question.

First: working through Lang's Basic Mathematics on your own and doing all the exercises is an impressive feat for someone with little formal education (in mathematics).

All four of the calculus texts you ask about are more difficult than the average text, but you should be able to manage any of them. Any of them will give you a good foundation for further study.

I know Courant and Spivak reasonably well, Apostol and Lang only by reputation and reviews. Courant is a classic. It will give you the best sense of the depth and usefulness of calculus, and how to think about mathematics. Spivak is probably the most thoughtfully rigorous. I think Apostol would be the most thorough, touching just about anything that might appear in any calculus curriculum. Lang will be straightforward, but not encyclopedic.

You don't say why you are in a hurry, or where you want to go next (more reading? back to school?).

I would suggest that you spend some time working through the first chapter or so of each - I think you can see that material on line. Then decide which suits your learning style best. You might want to study from two of the books, so you can compare the approaches and learn from two views.

Good luck.

Edit in response to the edit.

In six or seven months you should be able to prepare yourself for that exam.

Optimization might require some knowledge of multivariable calculus and linear algebra. That's in the second volume of Apostol, maybe a bit in Courant.

I don't usually recommend studying toward a particular exam, but if you can find old copies of the one you have to take you'll have a little more information on what to be sure to think about.

Is there someone at the school you can talk to now about optimizing your chances for admission?

I have a similar background most of my math has also been self studied. In fact I also started by reading Serge Lang's Basic Mathematics. This gives a great foundation and gives you a lot of nitty gritty stuff that will help you prove things later! If you can handle that book I definitely think you'll be fine reading his calculus book (I also read that book after reading Basic Mathematics).

Being a fellow self studier I highly recommend that you look at MIT open courseware. MIT posts top quality courses with lecture videos, problem sets, exams, and solutions 100% for free online. Sometimes learning straight out of the book can be difficult and this will be very helpful!

Here is the calculus course

Here is the discrete math course

Watching these lectures with the book will definitely help you prepare for the exam. In addition I highly recommend 3B1B's animated short videos that help visualize the concepts from calculus they are amazing!

3B1B videos

Good Luck!