With a swivel-blade vegetable peeler, remove the outer skin of the daikon. Discard the skin. Continue to peel down the length of the vegetable, removing the daikon in long, narrow ribbons, which look like noodles. Soak the "fettuccine" in a bowl of cold salted water for 15 to 20 minutes.
Meanwhile, make the tomato-basil sauce: Drain the tomatoes, reserving half of the juice. Squeeze the tomatoes through your fingers to mash them and combine with the juice; there will be about 2 cups.
In a heavy medium saucepan, heat the oil over medium-high heat. Add the onion and garlic and saute until softened but not browned, about 3 minutes. Add the tomatoes and their reserved juice, the sugar, and 1 teaspoon salt. Boil vigorously, stirring often, until the sauce is thick, 10 to 15 minutes. Stir in the basil and season with salt and pepper, to taste.
Drain the "noodles" and dry them on a kitchen towel. Add to the sauce and toss gently over medium heat, taking care not to break the "Daikon Fettuccine". Cook just until heated through, about 1 minute.
Divide among individual plates, teasing the fettuccine into mounds. Serve immediately.