In the bowl of a food processor fitted with the metal blade, pulse garlic until very finely chopped. Add walnut pieces and arugula; process until a coarse paste forms, about 5 seconds. Transfer to a serving bowl. Stir in the salt and Parmesan cheese, and set aside.
Bring a large pot of water to a boil. Add linguine, and cook until al dente according to package instructions, about 8 minutes. Drain in a colander, and immediately add to bowl with walnut-arugula mixture. Drizzle with the oil, and season with pepper. Toss thoroughly until coated evenly. Serve immediately.