masked man had the same price in mission dealer

That's not the definition of free. I'm pretty sure OP asked for time period, in which you can piece him up just from free points, free redeemables etc.

Mission dealer is by far the best "free" way of getting minato. You have 1050 free points, which translate into 10 free frags (11 if you spend 60cp for missing 30pts). Every other event, that lasts for a whole week allows you to redeem between 4-5frags - mostly balloon, time limited ninja recruitment (the one with daily keys from elite/wanted) or any other event, where you get points via sweeping instances. Now since ninjas rotate, we assume every other week theres event, where you get 5frags for free and every other mission dealer gives minato frags: thats 4x5 + 11=31 every 2 months. For completely full minato it would take about 5 months. You can speed up the process by buying minato frags in other events like fateful encounter or treasure of the sages, where he averages about 140-150cp/frag OR lucky board/mission dealer, where he costs about 180-200, but you get additional items from board/boxes. Those two events are the events you want to spend hoarded coupons anyway.