To start the Mesuking side-story, go outside the arcade on Nakamichi street and pick up the odd glowing butterfly.
Mesuking is a fighting card game that veterans will instantly recognize as a reskined JCC from 0. The fighting style is the same but instead of only choosing a girl and picking rock paper scissors at random you can actually pick what moves the buttons do. The cards are divided into two groups. The first are the "girls" themselves and the second are the moves they can do. Chapter 4 is the earliest you can start Mesuking and once you do the introductory fight with the professor, all of the remaining cards are scattered across town. below is a nice map and text guide for the location of the cards.
Location of all of the cards -
https://gamefaqs.gamespot.com/ps4/181171-yakuza-kiwami/faqs/...Rivals-
There are eight rivals needed to finish this side-story and get the achievement. to get more rivals just ask the professor and he will bring more. 7 of the 8 rivals are available in chapter 4 and the last is available in chapter 5.
#71- The professor will pair you with your first rival. The strategy to win is to just pick paper since he always does rock.
#72- The second rival can be instantly challenged after the first. The girl focuses on her best attack and not much else so just choose around that.
#73 - This rival will seem not able to be fought at first because he wants you to have a bee card in your deck. Fear not just go outside the arcade and you will see a card on the ground that just happens to be a wasp that will be approved for the kid. The strategy is to pick a rock based deck and just use rock on him.
#74- This is where you will see your first actual wall. The professor will only let you fight another opponent when you get 30 cards. Time to run around town collecting cards. Once you can fight, this opponent will lean towards scissors, but he does not seem to have a set pattern so it may take a couple tries to let rng work for you.
#75 - The next opponent has a pattern, but can decide to not use it entirely. His move set is just going counter-clockwise, so if he gets in this rut then just use the right moves according to his last. Sometimes, however, he will decide to not go with his normal strategy and pick seemingly random so just try your best to win.
#76 - The next rival is mainly paper based but has no set pattern.
#77 - The final rival for chapter 4 is scissor-based, and again has no set pattern.
#78 - The final rival requires you have at least 50 Mesuking cards which is not available until chapter five because you need to purchase them/ find them in purgatory. You can do this fight on the way to the batting center at the end of the chapter. The last rival does not have a set pattern so just defeat him using your skills and you will become the Mesuking King and the achievement will unlock.