Combine sugar, water, soy sauce, ginger, garlic, and salt in a small saucepan. Bring to a boil, stirring until sugar and salt are dissolved, about 5 minutes. Remove from heat and cool slightly, about 5 minutes. Stir in rice vinegar to make dressing.
Bring a large pot of water to a boil. Cook udon noodles in boiling water, stirring occasionally, until noodles are tender yet firm to the bite, 10 to 12 minutes. Drain. Rinse with cold water and drain.
Transfer udon to a large bowl. Add cucumbers, shallot, red bell pepper, and Thai chile pepper. Pour dressing into the bowl; toss until salad is evenly coated. Let stand to marinate, at least 20 minutes, before serving.