Mystic Art is the top-tier skill found in the relevant weapon skill-tree.
To be able to purchase it you need to meet the following criteria;
* Have over 500,000 proficiency points in the Sword (found under the Status tab > 2nd page)
* Have the necessary Samurai Points (7)
* Cleared the Dojo Mission 'The Way of the Warrior' (Veteran)
Once you have these done you can access Dojo Mission 'Zen and Sword are One' - a 1-on-1 battle with a sword-wielding master. He's lvl 100 so it's a challenging fight.
Beat him and you'll then be able to go into the skill-tree and spend those 7pts to unlock a Mystic Art of your choosing - the trophy unlocks straight away here. Thanks DrShow1976 for the details.