Preheat the oven to 325 degrees F (165 degrees C). Grease and flour a 10-inch Bundt pan. Sprinkle chopped walnuts evenly over the bottom of the pan.

Combine cake mix and pudding mix in a large bowl. Mix in eggs, 1/2 cup rum, 1/2 cup water, and oil until well-blended. Pour batter over chopped nuts in the pan.

Bake in the preheated oven until a toothpick inserted into cake comes out clean, about 1 hour.

Meanwhile, make the glaze: Combine sugar, butter, and remaining 1/4 cup water in a saucepan. Bring to a boil over medium heat; cook, stirring constantly, until it thickens and slightly darkens, about 5 minutes. Remove from heat and stir in remaining 1/2 cup rum.

Let baked cake sit in the pan for 10 minutes, then invert onto a serving plate. Brush glaze over top and sides. Allow cake to absorb glaze and repeat until all glaze is used.