Here is my full walkthrough of Act 4, complete with this achievement!
*Note: It has to be Tuesday to get this achievement. Play it Tuesday, or change your system to Tuesday*
1. After boating down the tunnel as Shannon and Conway, you'll go to a restaurant.
2. When you get to the top, go to the far right to the guy at the Claw Machine.
3. Use the following dialogue:
"Sure, I have one quarter"
"I'll try to get the headphones"
"Ezra pushes the joystick right"
"Ezra pushes the joystick up"
"Ezra pushes the joystick right"
"Ezra pushes the button to drop the claw"
4. On a Tuesday, you'll get the headphones, unlocking the achievement.