How to Get The Strength Achievement
You actually have to fight, and beat, all opponent in the boxing ring, and do the prison fluff piece.
On the 2nd sunday, you’ll be approached by a guard to go to the new route.
Be wary: You’ll have three (really) tough fights ahead of you without any weapon (but with time between each to use buff items, etc…), I recommend at least a NG+.
Then, once it is done, you’ll have sit up and escape when opportunity arise, or else you’ll be back in the prison.
Note that if you didn’t obtained the proof at the point, you won’t be able to get it before escaping.