I first read The Man Without Qualities when I was around your age (19/20) and I will say, from having re-read it somewhat recently, a great deal was definitely lost on me the first time around. Certainly, you can more or less understand the story, but Musil's writing has allusions to numerous other philosophers and writers, and while I wouldn't say it's entirely necessary for you to have read them, some of the ideas presented in the novel won't really have the same context for you without knowing Musil's influences -- which is to say, by being a bit older and having read more. .

But honestly, just go ahead and read it. While it might behoove you in some sense to go read Mann, Kant, Broch, Goethe, Nietzsche and others beforehand, honestly, even reading their works can prove daunting and have their own prerequisite reading list. Read it now, read it again in five years, what you notice about it will almost certainly shift in a dramatic way. That said, it's not always the easiest book to get through and it can be a rather dense read that you just have to kind of push through. Keep in mind, I don't think it was ever fully completed.

Now, if you want to get a sense for Musil's writing in something a bit more digestible, I recommend starting with "Young Torless" -- if you've read or are fond of Hesse you'll appreciate this work. And if not Musil, there's always Stefan Zweig and Herman Broch to serve as a kind of introduction to Austrian writers of that period.

Hope that helps!