It would take an
extraordinary effort. I doubt there is
anyone who could learn this material given anything less than a year of solid study -- and I am thinking of the leading researchers in the world, who are already experts in everything related. I am a professional mathematician myself, and I doubt I could manage it even within a year.
This is dictated by the difficulty of the subject, and it is even more so since Mochizuki's papers are notoriously difficult to read. What I have heard from other research mathematicians is that Mochizuki doesn't make much of an effort to make himself comprehensible, to answer questions, or in general to explain his results to the community.
Also please be warned that the consensus is generally that Mochizuki's theory is probably incorrect.
That said, if you want to learn the stuff, there is no reference other than Mochizuki's papers themselves. If you would like to learn some general background theory I would recommend learning some algebraic geometry, for which see Ravi Vakil's book:
http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pd...
and maybe Milne on etale cohmology:
http://www.jmilne.org/math/CourseNotes/LEC210.pdf
If you find that incomprehensible, start with Dummit and Foote's Abstract Algebra (read it cover to cover), Neukirch's Algebraic Number Theory, Atiyah and Macdonald's Commutative Algebra, and Fulton and Harris's Representation Theory.
If you indeed want to read all that, you may as well enroll in a Ph.D. program in math....