Preheat oven to 325 degrees F (165 degrees C). Grease a 12-cup fluted tube pan (such as Bundt(R)).
Beat white sugar, butter, milk, and vegetable oil together in a bowl using an electric mixer until smooth and creamy. Beat eggs, 1 at a time, into creamed butter mixture, beating well before adding the next egg.
Whisk flour, green tea powder, baking powder, and salt together in a separate bowl; slowly mix into creamed butter mixture until batter is smooth. Fold azuki beans into batter; pour into the prepare tube pan.
Bake in the preheated oven until a toothpick inserted into the cake comes out clean, about 70 minutes. Cool cake.
Invert cake onto a serving plate and dust with green tea powder and confectioners' sugar.