The Panic Puppet Zone boss involves Robotnik's Eggmobile being in a tube controlling 2 spiked arms and 2 side guns. To hurt Robotnik, you need to let the arm slam down to hit you, and dodge it. In the back next to the slammed arm's connecting point to Robotnik's cylinder is a blinking blue button. Hit it to damage him. You need to swipe around the edge or make a very precise hit on the spot. Stand in the middle to dodge the flying shots from the side guns. Once you hit one side a number of times, it will be destroyed. Destroy the other side and Robotnik will go down deeper into the cylinder. Follow him through the trap door.
Believe it or not...that was stage 1. Welcome to stage 2. Robotnik now has two flamethrowers that shoot homing fire blasts last seen in the Volcano Valley Zone boss. When he shoots a flame, the side he shot it from will have the same blinking blue button as the 1st stage. Hit it the same way as the 1st stage enough times and the arm will die the same as before. Destroy both arms to move lower.
Another stage?! Yep. This time Robotnik has rapid fire guns. These bullets can ricochet around the arena so be careful. Hit the blue buttons on either's cylinder hinges as before. The issue here is that the blue button is now only active for every bullet shot. Do this enough and the arm goes down. Do this to both arms to shatter Robotnik's cylinder and claim victory.