Historically speaking, perhaps an analysis course should begin with Fourier's approach to problems regarding the conduction of heat. (For more on this, see e.g. here.)
Of course, many math curricula are organized and re-organized (and re-re-organized) in ways differing radically from their material's original development. For example, Galois Theory is often offered as a sort of "Abstract Algebra II" course, though its creation (e.g. work by Galois himself) did not rely on the group theoretic arguments that you would now likely learn first in an "Abstract Algebra I" class.
As a matter of personal preference, I think it's helpful to start building a topological vocabulary in your first real analysis course (e.g. words such as those mentioned above by Robert Israel). One could argue that learning these definitions in a specific context will be confusing when moving to a more general context (e.g. what compact means in a real analysis course vs. what it will mean in a topology course). Others might argue that having specific examples will help when you go on to study objects of greater generality.
There are a lot of topics that could be covered in an introductory analysis course, and it's quite possible that what is deemed important enough for inclusion is related mostly to the instructor's background (e.g. someone with a background in harmonic analysis might take a more Fourieresque approach in lieu of introducing concepts from point set topology).
I cannot tell you what "should" be done, but I would include a portion on the standard topology in Euclidean space in my own course on analysis, partly because I think the terminology should be seen sooner rather than later, and partly because I find the material particularly interesting and fun to play around with.
I would recommend practicing bounding things with inequalities, so when you encounter a problem that requires the use of the triangle inequality (it comes up fairly frequently), you can find a bound fast. Many of the definitions used in analysis deal with inequalities as well, such as convergence of a sequence, limit of a function, continuity, etc...
Also maybe if you bought Rudin's book before next year and went through the definitions (such as in Chapter 2, where there is a lot of terminology/definitions on point set topology), it would certainly help, as when your professor teaches it next year, you will already have been accquainted. If Rudin is too difficult to read, perhaps try an easier read, such as Steven Lay's Analysis With an Introduction to Proof. I found it easier to read than Rudin. The key differences in terms of topics covered are that Rudin covers most of the material in metric spaces, while Lay focuses on $\mathbb{R}$ in particular. Also, Rudin discusses the Riemann-Stieltjes integral, while Lay sticks with the Riemann Integral.
Best Answer
They are both rigorous in that they both give complete proofs of their results. Rudin's problems on the other hand are challenging to newcomers. Abbot's problems are on a much lower level than Rudin's. I love Rudin's books, but there are mixed opinions on whether they should be used as introductions. I used Principles after my first year of analysis and loved it. I'd say first work through Abbot because he will likely provide more motivation. Later, get Rudin and push your boundaries of understanding. You might just become an analyst after that approach. It's what happened to me.