I don't think you need much topology or analysis at all.
It is however very difficult to work through an advanced text on axiomatic set theory, like Kunen's Set Theory, without having the mathematical maturity of at least an advanced undergraduate student. So, without experience with mathematical rigour (like you'd usually learn in a first course on Topology, Analysis, Group Theory, Measure Theory, and so on), it may be hard to appreciate the subtleties of set theory (and set theory is filled to the brim with subtleties).
If you've never worked through a basic text on Analysis or on Topology prior to learning Set Theory, then I'd recommend doing that just for the sake of becoming mathematically mature.
I'm not aware of books only covering the absolute minimum in Topology or Analysis, since the minimum necessary for Set Theory is too little to write a book about.
In general, any undergraduate introduction to Topology or Analysis will suffice, but here are some specific references:
The most advanced topology needed is likely product topologies and compactness.
- Chapter 2 from Munkres' Topology will likely cover about 95% of the topology you need to understand Kunen's text.
- These notes by Allen Hatcher cover enough. And you likely won't need chapter 4.
As far as Analysis goes, if you know what a Cauchy sequence is, you're probably covered on that front as well. You can certainly ignore anything about power series or complex numbers.
- I recommend Terence Tao's Analysis I for someone interested in Set Theory. Up to Chapter 5 should be plenty for a bare minimum, but I'd say there's useful extra information in chapters 6 to 9 as well.
- The first four chapters of Rudin's Principles of Mathematical Analysis.