The Challenge "Mystery Man" is for defeating Q in the arcade mode. So first off, the requirements to meet up with Q in the Arcade Mode are,
1) You cannot lose a round at all out of all the matches you play.
2) Get a rating of at least a D on every stage or better.
3) Get 5 Super finishes.
4) Get 2 or more Special Points.
Let's get this done the easiest way possible for you all.
On the main menu, go to "Help & Options", then "Settings".
Change the difficulty to 1 Star. And scroll down to "Dip Switches." Turn off the Guard ability and now your opponents will no longer be able to block your attacks.
I chose Ken, only because he's the fastest of the Shoto characters.
With every opponent, Jump High Kick and then Sweep them with a Crouching High Kick. Repeat the process against the enemy and without them being able to block, it should be a cinch.
You should be able to get your super meter up pretty fast so when they get low on Health, remember to finish them off with a Super. (I did it at least every match, just to be sure that I had it done.)
Continue up past Match 7 or 8 and after finishing it, Q should appear on screen. Your new opponent has appeared. Continue with the same strategy and you should be fine. (Make sure you beat him though, because if you lose he won't re-appear and you won't get the achievement, unless you do the process all over again.)
Hope this helps you all!
Helping credit to Kiplerr.