All right, Real Analysis is a proof-based course. If you have not had one of those before, it will be quite different from your other math classes. No longer will the professor tell you how to solve problems and then ask you to solve them. Now you will have to prove that the techniques to solve those problems are actually valid. You should be familiar with proof by induction, proof by contradiction, and proof by contraposition.
The course will start out covering thing that you’ve known since elementary school, but have probably never studied rigorously before. For example, in the section about field axioms, you might be asked to formally prove that 0·1=0. You might have to rigorously define what it means for a real number to be “positive”, and prove that the positive numbers are closed under addition, multiplication, and division, but not subtraction.
From the list of topics you provide, it seems likely your course will not make you prove that such a thing as the real numbers actually exists. Instead it will implicitly take the stance, “If a complete ordered Archimedean field exists, these are the properties it must have.”
Most of the course will stem from a few major ideas. One of these is the least upper bound property, meaning every bounded set of real numbers has a least upper bound in the reals. Another is trichotomy, meaning for any two real numbers x and y, exactly one of “x=y”, “x<y”, and “x>y” is true.
There will be a lot of topics dealing with limits from the ε-δ definition. This will probably be couched in the language of neighborhoods and balls, and likely will constitute your first introduction to the study of metric spaces. The course will start slowly, then progress quickly, until at the end you will be rigorously proving calculus theorems that you might not have seen before.
As far as preparation goes, it will help if you are able to read and write mathematical proofs. You should be able to follow formal logical reasoning, and preferably able to produce your own. There’s not much in the way of “math” background that you’ll need. The course is mostly self-contained, and will build things up from basic arithmetic all the way to calculus, proving almost everything along the way.