Prepare the soup. Heat dashi to a boil and add salt, soy sauce, mirin, and sake. Let cool first, then chill the soup in the fridge until cold.
Prepare the "Kitsune" topping. Cut aburaage (thin tofu) into 1/2" (1 cm) strips. In a pot, combine water, soy sauce, sake, mirin, and sugar, then cook over medium heat and add the prepared tofu. Cook until most of the liquid has evaporated. Let cool and chill.
Once the other components have become cold, prepare the udon noodles. Cook the noodles according to package directions, then cool under running water, drain, and cover with ice to chill.
Divide the cold noodles and soup into shallow bowls, place the cooked aburaage on top, and sprinkle with chopped green onions. Serve immediately.