The boss battle takes place inside Muggshot's casino. The room is filled with huge crystals and mirrors, which Sly must use to defeat Muggshot. The second floor is very similar to the first, and the last part of the boss battle takes place on the chandelier of the casino, hanging high above it all.
Muggshot will use his machine guns to shoot Sly if he sees him. That's the only attack Muggshot uses, however, if he manages to hit Sly, he will die unless he's carrying a Lucky Charm.
Avoid Muggshot and turn all the mirrors without getting hit. Once all the mirrors are turned towards the crystals, Muggshot will get burned by the sunlight and his weapons will be destroyed. He will then move on to the second floor, where you'll have to repeat the process. Using the Roll move, Muggshot won't be able to hit you, allowing you to hit the mirrors without having to worry about dying.
The third stage of this boss fight is a bit trickier. Muggshot will be surrounded by mirrors, and you'll have to wait until he shoots one to actually turn the mirror around. Once you've managed to successfully turn the mirror, quickly move on to the next. Time your moves correctly according to Muggshot's behavior, because if you don't, you'll either die and start the boss battle all over again, or have to turn all the mirrors again. Once all the mirrors have been properly turned around, Muggshot will be defeated, and Sly will recover the second part of the Thievius Raccoonus.