Put a large pot of water on to boil and salt it. Stir together the dashi, soy sauce, vinegar, sugar, sesame oil, and fresh ginger in a small bowl. Taste, add more soy sauce if necessary, and let sit. Dissolve the wasabi in the 1/4 cup boiling water; let cool and thicken for a few minutes. Add a little more powder if it seems too thin.
Drop the ramen into the pot of boiling water and cook, stirring occasionally, until tender but not mushy. Start tasting after 1 minute; the cooking time can vary by a few minutes. Drain, rinse in a colander under cold running water, and drain again.
For individual servings, divide the noodles among 4 serving bowls, then divvy up the cucumber, nori, and pickled ginger among them. (In Japan, these ingredients are usually nested in individual piles on top of or around the noodles, though you can just scatter the toppings over the noodles if that seems too fussy.) Top each salad with a pinch toasted sesame seeds and the scallions.
Serve the noodles with small bowls of the dressing on the side for diners to dress and toss the salad at the table. Pass wasabi to season the noodles.