Preheat oven to 325F. Prepare a 12-inch Bundt pan by spraying liberally with nonstick baking spray.
In a stand mixer fitted with paddle attachment, combine butter, oil, pudding mix, sugars, baking powder, and salt and beat on high for 3 minutes, until evenly mixed and fully hydrated.
Reduce speed and add flour in 2 additions and beat until thick and smooth.
Add eggs one at a time, scraping down the bowl between each egg.
Add vanilla, milk, and rum, and mix until smooth. Batter will be thinnish.
Pour batter into prepared pan. Bake until golden brown and cake starts pulling away from the edge of the pan, about 50-55 minutes.
Remove from oven and let cool in pan for 15 minutes.
Meanwhile, combine all the syrup ingredients in a small saucepan. Heat over medium heat and cook until the mixture thickens and reduces, about 6-7 minutes.
Using a skewer, poke holes all around cake. Slowly pour the syrup all over the bottom of the cake, allowing it to soak up. Cover and let sit for at least 2 hours to fully soak.
Turn out cake onto platter and serve.