If you're concerned about time, I don't think reading Calculus by Spivak is the best thing to do.
Either Zorich or Apostol is a great choice. I would say that they're "intermediate" in difficulty. Zorich contains more in-depth discussions of topics, and more examples than does Apostol.
If your goal is only to move on to Royden, you'll probably cover the material more quickly in Apostol. Zorich covers a number of topics not addressed in Apostol, such as vector analysis and submanifolds of $\mathbf{R}^n$. These are important topics, but not direct prerequisites for Royden. Still, I think with all the Lagrange multipliers and similar tools people use in economics, the submanifold topic is important if you want to understand the theory very clearly.
Zorich's first volume is quite concrete, whereas Apostol becomes abstract more quickly. This is probably because he doesn't want to duplicate what would be in a rigorous calculus book like his Calculus, although he does this more than Rudin's book does. His analysis book was for second- or third-year North American students, whereas Zorich's is, at the outset, for first-year Russian ones. Russian students have typically had some calculus in high school, but the practical portion of learning calculus continues into their first-year of university, with harder problems. So in Zorich I, you deal with hard problems on real numbers, rather than delving straight into metric spaces as you would in Apsotol's book.
Zorich covers only Riemann integration, whereas Apostol has chapters on Riemann-Stieltjes integration in one variable, Lebesgue integrals on the line, multiple Riemann integrals, and multiple Lebesgue integrals. The treatment of Lebesgue integration is less abstract than in more advanced books. Since it's limited to $\mathbf{R}$ or $\mathbf{R}^n$, it's more elementary, but at the same time there is some loss in clarity compared to the abstract theory on measure spaces. One reason to use Apostol might be a sort of introduction to the Lebesgue theory before returning to it at a higher level and "relearning" certain parts of it. Whether you'd want this is up to you.
The fact that both Rudin and Apostol have chapters on Riemann-Stieltjes, rather than Riemann, integration, indicates to me that they assumed students had already studied Riemann integrals rigorously, and would be ready for a generalized version right from the start. Considering the type of calculus courses most students take these days, this is rarely the case now. Zorich doesn't have this problem.
All in all, for a typical student who is good at math but didn't learn their calculus from a book like Spivak's or Apostol's Calculus, I think Zorich is the better choice because of the more concrete approach in the first volume (this doesn't necessarily mean easier). On the other hand, time constraints might cause you to prefer Apostol's analysis book.
EDIT: An important point that I neglected to mention is that Zorich's book will be much better than Apostol's if you aren't yet acquainted at all with multivariable calculus. A practical knowledge of some multivariable calculus is probably one of the tacit assumptions that Apostol and Rudin make about their readers, which is what allows them to deal with multivariable calculus in a briefer and more abstract way. Compare Apostol's 23-page chapter on multivariable differential calculus to Zorich's 132 (in the Russian version).
EDIT: Based on your later comments, I would suggest that reading
Spivak's Calculus,
Whichever you prefer of Apostol's Mathematical Analysis or Rudin's Principles of Mathematical Analysis.
would be a reasonable plan.
However, before beginning the multivariable calculus parts of those books, it would be best to learn some linear algebra and multivariable calculus from another source. This could be Volume 2 of Apostol's Calculus. You could instead skip straight to the multivariable part of Volume 1 of Zorich, but you'd have to learn the necessary linear algebra elsewhere first. I don't recommend Spivak's Calculus on Manifolds if you want to learn multivariable calculus for the first time. Also, you won't need Munkres - you'll get enough topology to start with in whichever other book you read.
EDIT: In answer to your additional question, these topics are mostly not discussed in Spivak.
However, Spivak is an excellent introduction to the mathematical way of thinking. That is, although you will not learn all the specific facts that arise in higher-level books (you do learn many, of course), you will learn to read and understand definitions, theorems and proofs the way mathematicians do, and to produce your own proofs. You will become intimately familiar with real numbers, sequences of real numbers, functions of a real variable and limits, so you will have examples in mind for the more general structures introduced in topology. You will also solve difficult problems.
So it is not that you will know topology already when you've read Spivak's book, it's mainly that it ought to be easier for you to learn because you will have improved your way of approaching mathematical questions. Countable sets are in fact discussed in the exercises to Spivak, however.
I can't guarantee that your trouble will "go away," but there is a good chance it will.
Also feel free to use Zorich rather than Rudin or Apostol, after Spivak, or even to jump straight to the multivariable part of Zorich at the end of Volume 1 and start reading from there.
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?
Best Answer
First of all, the writing style of Baby Rudin is very concise and it's difficult to understand Rudin's Mathematical Analysis just by reading it one or two times without previous exposure to the topic. Therefore, I advise you to stay away from that book at all costs. There are better books available these days that have been written better and offer more insights and intuition about introductory Analysis such as Pugh's Mathematical Analysis.
One particular area that Rudin's Mathematical Analysis hasn't covered well is the equivalence of sequential compactness and compactness by open covers. It hasn't talked about the Lebesgue number of an open cover at all. But that's not all of it. You want to study financial mathematics, that means most importantly you need to learn measure theory which is covered in the last chapter of Baby Rudin and it is so abstract that when you read it for the first time, you will have absolutely no idea what he is talking about. Pugh's Mathematical Analysis is more verbose and it develops your intuition when it discusses measure theory.
Also, to understand financial mathematics, I assume you have to learn about stochastic integration and Ito's formula and maybe even Malliavin Calculus. This requires a fair amount of knowledge in "Real Analysis", not "Mathematical Analysis". There are topics that are not covered in a Mathematical Analysis course such as signed measures and Radon-Nikodym Theorem.
I believe Spivak's style of writing is more geometric than Apostol in general, therefore, I believe it is better if you want to cover the topics necessary for future understanding of differential manifolds.
However, I advise you to choose Apostol's Calculus over Spivak and then continue to read Apostol's Mathematical Analysis. It is less difficult to read than Baby Rudin, it is verbose and covers a lot more details than Baby Rudin, it covers Bounded Variation functions that Baby Rudin does not cover, and it discusses Riemann-Stieltjes integral in such a detail that is way beyond someone's knowledge who has studied only Baby Rudin. It also discusses functional spaces in more details than Baby Rudin.
Therefore, my suggestion is like this: 1- Apostol's Calculus, Volume 1. 2- Apostol's Mathematical Analysis, excluding the chapters related to multi-variable analysis or Pugh's Mathematical Analysis excluding the chapter related to multi-variable calculus.
Also, I advise you to review Calculus by solving like $10\%$-$20\%$ of the problems in Apostol's Calculus, volume 1. Having learned Calculus before, and exposure to its ideas and intuitions is definitely helpful, but do not spend too much of your time (like more than $2$ weeks) on Calculus because it's really not that necessary for learning Mathematical Analysis.