Bring a large pot of salted water to a boil over high heat. Add the linguine and cook until it is al dente, stirring frequently, about 8 minutes. Drain, reserving 1 cup of the cooking liquid.
Immediately toss the hot pasta in a large bowl with the butter to coat. While tossing the linguine, gradually sprinkle the cheese, pepper, and enough of the reserved cooking liquid evenly over the linguine to moisten. Add the arugula and toss to combine. Season the pasta, to taste, with salt. Divide the pasta among 6 plates and serve.