The Shady Guy is found in two places in A Link Between Worlds. He is first encountered right outside of Zora's Domain after Link discovers that the Smooth Gem has been stolen. He is next encountered in Kakariko Village, where he stays for the remainder of the game.
The Shady Guy steals the Smooth Gem, which sets Link off on a quest to find and return it, allowing him to receive the Zora's Flippers. The Shady Guy also holds the Pegasus Boots. Link can get these from him if he sneaks behind him while in drawing form and pop out on top of him.