You definitely need to know some grammar before tackling a book. I’ve not actually heard of that resource you linked to, so can’t comment on that. Personally, I worked my way through the Genki textbooks. If you’re looking for free resources though, there are things like Tae Kim’s Guide and the Cure Dolly YouTube channel (it’s the most horrific thing you’ve ever listened to, but advocates swear you get used to it / recommend speeding up the video / turning off the sound altogether and using captions).
How long it will take before you can read that book will depend on how much time you have available to study, and how quickly you learn we put together a vocabulary sheet as you go, so grammar is actually much more important than vocabulary in this case (it doesn’t use many kanji either).
If you’re really determined to set that as your first reading goal, you could consider buying it, and then periodically having a go at reading the first page or two to see whether you’re ready to tackle it yet!
If you’re interested in reading right away, graded readers might be a better start. They’re intended for foreigners and they start off with very basic grammar, so you could get started without knowing too much grammar. I think there’s a thread floating around somewhere about graded readers that includes some free PDFs. You could read them on the computer or print them off into a book. The idea is to read them without a dictionary and learn what the words mean from the pictures and the context.