I would recommend you to read the first several pages of chapter 0 (up to cell complexes) in order to get some intuition for homotopy and then go directly to chapter 1. Then you can return to chapter 0 to understand cell complexes and other definitions when you encounter them in other chapters. I also studied algebraic topology via this book and it is indeed possible to cover chapters 1 & 2 within 2-3 months of intense work.
I would like to say that Hatcher's book is very rigorous and the everything is given in very technical, algebraically precise way, but sometimes it lacks intution and there are very few pictures so it might be hard to read for someone who is new to topology. So I would recommend you to read Massey's: Algebraic Topology: An Introduction which has many more pictures and explains things in a more geometrically intuitive way. In my opinion it is a much better introduction.