Cook noodles according to package directions. Drain, rinse with cold water, and drain again thoroughly.
Bring 3 cups water, sake, and ginger to a boil in a Dutch oven or stockpot over high heat. Reduce heat to medium-low to maintain a gentle simmer.
Peel daikon and cut into 4 6-by-3-by-1-inch slabs (about 8 1/2 oz. each). Cut each slab into 2 3-inch squares. Place in sake mixture; simmer until just tender, 10 to 12 minutes. Drain and dry well on paper towels.
Stir together honey, tamari, vinegar, and crushed red pepper. Set aside 1/4 cup of the honey mixture.
Heat oil in a large nonstick skillet over medium-high heat. Cook daikon squares in batches until lightly caramelized, 3 to 4 minutes per side. Reduce heat to medium. Turn daikon and brush with 2 tablespoons of the honey mixture. Cook, turning and brushing often with honey mixture, until deeply caramelized, 5 to 6 minutes. (Watch carefully to prevent glaze from burning.)
Toss together noodles, bok choy, almonds, and reserved 1/4 cup honey mixture in a large bowl. Divide noodle mixture evenly among 4 shallow bowls. Top each with 2 daikon pieces.