How long do I study on my own before applying to math grad school? I'm one of those who realized near the end of their Bachelor's in Business Mgmt that they really really love math. I'm interested in research in pure, theoretical math. I am confident in my ability to learn advanced math on my own. I admit this is based on a small sample of experiences I've had so far with self-studying. After a year of working, I decide to do it full time.

Since I'm behind my peers, I was thinking to just learn the fundamentals (Logic, Real Analysis, Abstract Algebra and Geometry) on my own, and then apply for a Master's. The reason I want to do these on my own is that I find I understand things better at my own pace taking the time to solidify the fundamentals. I took a year of math courses on an exchange program at London School of Econ and it felt like learning disconnected facts. Something like saying object x is round, black, with 4 holes etc. vs showing you a picture a black shirt button (which I find more efficient mentally). Concepts were introduced which were too new but no time was given to "form a picture" of the objects. As a result, it felt like very "syntactic". If I understand the basic concepts used over and over in math, rather than things built out of those facts like calculus, linear algebra, applied math and other "less abstract" branches, I feel I'd do well in my Master's course. I love taking the time explore mathematical concepts on my own, seeing how things fit. I find quite often I make my own concepts (simple ones though) where the picture feels incomplete, which is why I want to do research. 

I was wondering how long I should continue this approach (i.e. when have I learned enough fundamentals)? More objectively, what should aim to master on my own, before applying to a master's program? (unless you disagree this is the right approach)

PS: I'm 22, and I have financial support at the moment while I get this done. I've started with Logic and Set Theory and becoming good at proofs. I find calculus and linear algebra intro books skip over too many of the more abstract underlying concepts, hence the bottom up approach.
I disagree that this is the right approach. Many grad schools in the US would require you to have a Bachelor's degree in some subject and to make up the courses you missed by not having a Bachelor's degree in math. So your best option is to get started actually completing those courses. I think that unless you have some publications in math with someone who can vouch for your skills even though you don't have the coursework, you're not going to get into a good Master's degree program without the necessary coursework. The further your history is from the required classes, the more work you're going to need to do. Some programs might admit you with the requirement that you take a few semesters of undergrad classes to catch up, but if you need more than a few catch-up courses, you're really looking at a second Bachelor's degree or working on the side through a non-degree-seeking program at your local university.