Chapter 1 of Hatcher corresponds to chapter 9 of Munkres. These topology video lectures (syllabus here) do chapters 2, 3 & 4 (topological space in terms of open sets, relating this to neighbourhoods, closed sets, limit points, interior, exterior, closure, boundary, denseness, base, subbase, constructions [subspace, product space, quotient space], continuity, connectedness, compactness, metric spaces, countability & separation) of Munkres before going on to do 9 straight away so you could take this as a guide to what you need to know from Munkres before doing Hatcher, however if you actually look at the subject you'll see chapter 4 of Munkres (questions of countability, separability, regularity & normality of spaces etc...) don't really appear in Hatcher apart from things on Hausdorff spaces which appear only as part of some exercises or in a few concepts tied up with manifolds (in other words, these concepts may be being implicitly assumed). Thus basing our judgement off of this we see that the first chapter of Naber is sufficient on these grounds... However you'd need the first 4 chapters of Lee's book to get this material in, & then skip to chapter 7 (with 5 & 6 of Lee relating to chapter 2 of Hatcher).
There's a crazy amount of abstract algebra involved in this subject (an introduction to which you'll find after lecture 25 in here) so I'd be equally worried about that if I didn't know much algebra.
These video lectures (syllabus here) follow Hatcher & I found the very little I've seen useful mainly for the motivation the guy gives. If you download the files & use a program like IrfanView to view the pictures as you watch the video on vlc player or whatever it's much more bearable since you can freeze the position of the screen on the board as you scroll through 200 + pictures.
I wouldn't recommend you treat point set topology as something one could just rush through, I did & suffered very badly for it...
You didn't exactly ask "what background do I need to learn HoTT", but since that's the question some other people are answering, I'll address that too. The subject as a whole is quite wide, and if to understand it all and its applications deeply would require significant background in homotopy theory, higher category theory, topos theory, and type theory. However, none of that is necessarily required at the beginning, and indeed learning HoTT may help you get a handle on those other subjects at the same time or later on.
The book Homotopy type theory was written with the intent of assuming as few prerequisites as possible, not even basic algebraic topology or type theory, although it does assume some mathematical maturity and perhaps more category theory than would be ideal. If you don't have any exposure to category theory, I would recommend doing a bit of reading there; some good introductory books are Awodey's Category theory and Leinster's Basic category theory. But other than that, I would suggest diving into the HoTT Book and see how you find it. The first chapter of the HoTT Book is an introduction to type theory, intended to stand on its own; it's necessarily brief and only covers the basics necessary for the rest of the book, so you may want to supplement it with other readings, but unfortunately really good introductions to type theory are hard to find.
The HoTT Book is only about one particular facet of the subject, namely developing mathematics internally to HoTT. If you want to learn about semantics, for instance, you'll have to go elsewhere, and eventually need more background in abstract homotopy theory and higher category theory. May's book is not an unreasonable place to go after Hatcher; particularly because May uses the modern language of category theory (which Hatcher seems unaccountably allergic to), which is also at the basis of HoTT. Beware that Concise lives up to its name and requires a lot of the reader. The sequel More concise algebraic topology is mostly not hugely relevant for HoTT, but it does contain a good introduction to model categories, which you'll have to learn about eventually if you want to understand the semantics of HoTT.
Finally, as I mentioned in the comments, I also suggest availing yourself of the other resources linked at the homotopytypetheory.org web site, which include several video lectures and basic introductions written for people with differing backgrounds.
Best Answer
One of the classic references to studying algebraic topology is Hatcher's Algebraic Topology, which is available online at Hatcher's webpage. He says the following on the topic of prerequisites:
You should probably study the following collection of topics: topological spaces, continuous maps, connectedness, compactness, separation, function spaces, metrization, embedding theorems, and the fundamental group. You should also know what is taught in a "standard undergraduate course in algebra". A nice collection of notes written by Professor Richard Elman is here.
Since you say that you want to study algebraic topology with a "homotopical viewpoint", you should check out this MathOverflow question. Note: you can actually study homotopy theory through combinatorial objects called simplicial sets, but it's probably better to learn homotopy theory the classic way.
Hope this helps.