You can find the Coin Case two buildings east of the Game Corner. Once you enter the building, talk to Mr. Crowley on the left and he will give you the Coin Case. After that you can go back to the Game Corner and start playing the slots!
The Coin Case in the Johto region can be found in the Goldenrod Underground. Once you grab it, head on over to the Game Corner to start playing the slots and winning big!