Instructions
Go to Goldenrod City and defeat the Pokemon Gym Leader, Whitney. All of the Pokemon in this gym are Normal-type, so battle them with a Fighting-type Pokemon.
Leave the Gym after getting the Plain Badge for beating Whitney. Enter the house that is right next to the Gym in the northeastern corner of the town.
Talk to the girl in the house, and she will give you a Squirtbottle.
Head north through the National Park to Route 36. You'll encounter a shaking tree blocking the path; this is Sudowoodo.
Go up to Sudowoodo and press "A." You will get a message reading, "It's a weird tree. Use Squirtbottle?" Select "Yes."
Battle Sudowoodo with a Water, Grass, Fighting or Steel-type Pokemon. Wear down Sudowoodo's health, and then catch it with a Poke Ball. This is the only chance you'll get to catch a Sudowoodo in the game, so be careful not to knock it out.