Position a rack in the middle of the oven. Preheat the oven to 350°F Line 12 muffin tin cups with paper cupcake liners.
Make the swirl and topping: In a small bowl, stir the sugar and cinnamon together.
Make the cupcakes: In a small bowl, mix together the grated apple and the 2 tablespoons cinnamon sugar; set aside.
In a large bowl, stir the flour, sugar, baking soda, and salt together. Make a well in the center of the flour mixture and add the eggs, oil, and vanilla, stirring until blended. Stir the reserved apple mixture, with any juice that bas been released.
Spoon about 2 tablespoons of the batter into each paper liner. Sprinkle ½ teaspoon cinnamon sugar over the batter in each liner. Spoon the remaining batter over the cinnamon sugar, using a scant 2 tablespoons of batter for each cupcake.
Use a pastry brush to dab the top of the batter with the melted butter. Sprinkle the remaining cinnamon sugar over the top of the cupcakes.
Bake just until the top feels firm and a toothpick inserted in the center comes out clean, about 25 minutes.
Cool the cupcakes for 10 minutes in the pan on a wire rack. Remove the cupcakes from the pan onto a wire rack and let cool completely.