How long would it take someone like me, who can institutionally have basic conversations, to learn grammar and vocab on an academic level?
Hard to say. I suspect that if you have the 'feeling' of the language, it should be relatively easy. I think you could try reading a book on a subject that you're interested in and see yourself how it goes.
If you find that you only need a dictionary than it may take just a few months to build your vocabulary. But if you find that you have also problems understanding the grammar then you should consider taking a language course.