1. Wipe kombu clean with a cloth; cut sheeting lengthwise into 3-inch pieces. Place kombu in a large saucepan over medium-high heat; cover with 2 quarts cold water. Heat mixture to 180° or until tiny bubbles form around edge (do not boil). Reduce the heat to medium-low, and simmer 20 minutes. Remove from heat; discard kombu. Stir in bonito and ginger; let stand 10 minutes or until bonito sinks. Strain through a cheesecloth-lined colander over a bowl; discard solids.
2. Wipe pan clean. Return broth to pan; bring to a boil. Remove from heat. Stir in mushrooms; let stand 20 minutes. Strain mixture through a cheesecloth-lined colander over a bowl, reserving mushrooms and broth. Wipe pan clean. Return broth to pan; bring to a simmer. Coarsely chop mushrooms.
3. Cook noodles according to package directions; drain. Cut onions into 1-inch pieces; cut each piece in half lengthwise. Cut onion pieces lengthwise into thin strips. Combine sugar, soy sauce, and vinegar in a small bowl, stirring well. Add shrimp to broth; cook 1 minute. Stir in mushrooms; cook 30 seconds. Remove from heat; stir in vinegar mixture. Place about 1 cup noodles in each of 4 bowls; top with 2 cups broth mixture. Divide onion strips evenly among bowls.