Preheat the oven to 325°F.
Place the flour, pudding mix, apple sauce, sugar, baking powder, salt, corn oil and butter in a mixing bowl, and mix at medium speed until everything is thoroughly combined and the mixture looks like wet sand.
Beat in the milk, then beat in the eggs one at a time. Scrape the bowl thoroughly to ensure everything is mixed.
Stir in the rum, vanilla, and rum extract, if using.
Spray a 10- to 12-cup non-stick Bundt pan with cooking spray. Add a bit of flour into the pan and turn to coat thoroughy. This will help prevent the cake from sticking to the pan. Pour the batter into the prepared pan and spread it evenly with a spatula.
Bake the cake for 50 to 60 minutes. When done a long toothpick or wooden skewer will come out clean when inserted into the center. Remove the cake from the oven and leave it to cool for a bit while you make the syrup.
In a saucepan, combine the syrup ingredients, except the vanilla. Bring to a rapid boil then reduce to a simmer and cook (without stirring) for about 5 to 8 minutes, until the syrup thickens slightly. Remove from the heat and stir in the vanilla.
Use a long skewer to poke holes all over the cake. Pour about 1/4 cup of the syrup over the cake (still in the pan). Allow the syrup to soak in, then repeat again and again until all the syrup is used.
Cover the pan loosely with plastic wrap and allow the cake to sit overnight at room temperature to cool completely and soak in the syrup. When ready to serve, loosen the edges of the cake and invert onto your serving plate. If the cake won’t release, don't force it. Place it in the oven, turn the oven to 350°F, and warm for about 10 minutes, to soften the sticky syrup. (If your oven is one that preheats by making its upper element red-hot, place the cake on a lower rack and tent it with aluminum foil to protect it.) Remove the cake from the oven, and tip it onto the serving plate.
Wrap securely (or place under a cake cover) and store at room temperature for several days. Freeze for longer storage, up to 1 month.
Using the full amount of syrup in the recipe will make a traditional rum-soaked rum cake, one that's incredibly moist. For a less "rummy" cake, one with a more standard cake texture, make and use just half the amount of syrup.
Want to make a non-alcoholic version of this cake? Here’s how:
For cake with rich, buttery, vanilla flavor and undertones of rum — Cake: Substitute water for the rum, and use 1 teaspoon rum extract. Syrup: Substitute water for the rum, and add 1/2 teaspoon rum extract when you add the vanilla.