From the cover text of How to Build a Brain it seems the main focus is on the architecture of SPAWN, and I suspect it does not actually give a proper introduction to other areas of computational neuroscience. That said, I wouldn't be surprised if it is the most enjoyable book to read on the topic, that you can find. I have read Computational Neuroscience by Hanspeter Mallot, which is very short, weird and not very good. I'm currently about halfway through Theoretical Neuroscience by Dayan and Abbott. My impression is, it might be decent for people with a strong physics/math background, it's OK if you have some prior knowledge about the topics (e.g. having visited a lecture) and rather bad otherwise.

Edit: My prof told me about Information Theory, Inference and Learning Algorithms (legal free online version), which is, as the title implies more about information theory and learning algorithms (so more mathy), but from the perspective of neuroscience, so it's missing a lot of the typical topics of computational neuroscience. I have just started reading it, but so far it seems really well written (4.35 rating on goodreads), and it also contains exercises and reflection questions.