Preheat oven to 350*F. Brush or wipe bundt pan with oil making sure to get into all the crevices so that the cake will come out easily. Dust with flour and remove excess.
Sprinkle nuts in the bottom of the pan and set aside.
Place the cake mix, pudding mix, rum, oil, water and eggs in a large mixing bowl. Blend with an electric mixer on low speed for 1 minute. Stop the machine and scrape down sides. Increase the mixer speed to medium and beat 2 minutes more. The batter should look thick and smooth. Pour the batter into the prepared pan, smoothing out the top with a rubber spatula.
Bake the cake until it is golden brown and springs back when lightly pressed with your finger, 55 to 60 minutes. Remove the pan from the oven and place it on a wire rack to cool for 20 minutes. Run a long knife along the edges of the cake and invert on a serving plate. Poke holes in the top of the cake with a wooden skewer.
Prepare glaze: Place butter in a small saucepan and melt it over low heat, 2 to 3 minutes. add the water and sugar, stirring. Increase the heat to medium and bring the mixture to a boil. Reduce the heat slightly and let the glaze simmer until thickened, 4 to 5 minutes, stirring constantly. Remove the pan from the heat and stir in the rum. Spoon the glaze over the warm cake, allowing it to seep into the holes and drizzle down the sides and into the center of the cake. Allow cake to cool completely before slicing.
Store the cake covered in foil or plastic wrap, at room temperature for up to 4 days or in the refrigerator for up to 1 week.