Mystic Art is the top-tier skill found in the relevant weapon skill-tree.
The Axe is slightly different to the other 4 weapons, in that it has a side mission requirement as well, to be able to purchase the Mystic Art you need to meet the following criteria;
* Have over 500,000 proficiency points in the Axe (found under the Status tab > 2nd page)
* Have the necessary Samurai Points (7)
* Done the Kinki Region sub-mission 'Greater Demon Hunting' (lvl 70)
* Cleared the Dojo Mission 'The Way of the Warrior' (Veteran)
'Greater Demon Hunting' sees you fight a familiar Yokai, except the demon is now a lot faster and has a few ridiculous 1-hit kill attacks (some are over 3500+, so be warned).
During the battle a Samurai will leap in to help you - he's not that great but he acts as a decoy so use this window to heal-up, use Powers or prepare Talismans etc.
I cleared this on my 4th attempt (I was lvl 115) and the Samurai had to be assisted every time.
Once you have these done you can access Dojo Mission 'The Invincible' - a 1-on-1 battle with the Axe-wielding master who fought alongside you earlier. He's lvl 100 so it's a challenging fight.
Your opponent is right handed, so all his attacks are going to be led by that, and all come from that side - for an easy win all you have to do is dodge to YOUR right and you'll be able to land a strike or two - rinse and repeat for an easy victory.
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.