Adjust oven rack to middle position and preheat oven to 350°F. Spray a bundt pan with non-stick spray.
In medium saucepan, bring chopped apple and cider to boil over medium-high heat. Reduce heat to medium and simmer until most of the cider has been absorbed and apples are easily smashed with a fork, 10 to 12 minutes. Remove pan from heat, cool 5 minutes, and mash with the back of a spoon. Measure out 1 cup apple mixture and stir in milk; set aside.
In medium bowl, whisk together flour, baking powder, baking soda, salt, nutmeg, and mace; set aside.
In large bowl, beat butter, sugar, and brown sugar on medium speed until light and fluffy, about 3 minutes. Add eggs, one at a time, beating well after each addition. Add oil and beat until incorporated, about 1 minute.
Decrease mixer speed to low and add flour mixture in three batches, alternating with apple mixture, scraping down sides and bottom of bowl with rubber spatula as needed. Increase speed to medium and beat mixture just until combined, about 20 seconds. Add vanilla and beat once more, just to combine, about 10 seconds.
Scrape batter into prepared pan. Bake until cake tester inserted in cake comes out clean, rotating cake halfway through baking, 35 to 45 minutes. Transfer cake to cooling rack set inside baking sheet and cool in pan 10 minutes, then invert directly onto cooling rack.
For the Cinnamon Sugar Coating: Combine sugar, cinnamon, nutmeg, and salt in small bowl. Sprinkle warm cake with cinnamon sugar, using fingers to rub it onto sides.
Cool cake completely, about 1 hour.