It's really not as bad as it looks. The context dependence might be weird at first, but it works in your favor too. Nips like to say the bare minimum to make themselves understood, completely omitting anything that's already implied by the context. This tends to make sentences short and simple.
As for how long it takes to develop a decent parsing speed, it's really hard to give a satisfactory answer to that question because of how complex and varied the process of learning and the language itself can be, but if you learn some basic grammar (couple of months at a very chill pace), then pick up any LN or VN and plough your way through it like a mad man (anything from a couple of months to a year depending on the length of what you're reading and how much time you dedicate to it), you'll have seen most common sentence patterns countless times in several different contexts and a lot of it will have become second nature to you by the time you're done. It will be painful, but if you repeat that a handful of times (next time around it will be much faster cause you'll be a faster reader) you'll be reading pretty fucking well in no time.
If you don't want to brute force your way like that and would rather take it easy at first, you can still definitely get to a level where you could read something easy like SoL manga at a comfortable pace (not counting the time spent reading the definitions of words you don't know) in less than a year if you put your mind into it, i.e. practice reading every day and study grammar.
tl;dr read a lot and shit will start making sense before you know it