The absolute fastest way to PR is through Highly Skilled Foreign Professional Type 1 visa. PR after a year. That's incredibly fast. Type 2 visa after 3 years IIRC. The difference between 1 and 2 is the number of points one scores.
It seems that if you have a master's degree from top 300 universities, passed N2, aged under 30 and have high salary (10M yen+) you can easily get 100 points. Type 1 visa only requires 80 points and guarantees permanent residency in one year.
For reference, believe that's ~ USD$91k, EUR€82k, GBP£72k.