Preheat oven to 400 degrees F (200 degrees C). Grease a deep 4-quart baking dish.
Beat 1/2 cup butter and 1 cup sugar together in a large bowl with an electric mixer until light and fluffy. Stir eggs into butter mixture until smooth; stir in grated apple.
Whisk flour, baking soda, cinnamon, 1 teaspoon nutmeg, and salt together. Stir flour mixture into apple mixture; spoon into prepared dish.
Bake in the preheated oven until a toothpick inserted near the center comes out clean, about 35 minutes.
Heat remaining 1 cup sugar, remaining 1/2 cup butter, and milk in a small saucepan over low heat until sugar is melted, about 10 minutes; stir in rum and remaining pinch nutmeg just before serving. Serve sauce over pudding.