In a large pot of boiling, salted water, cook the linguine for 5 minutes less than package directions specify. Drain, reserving 2 cups cooking water.
Meanwhile, in a large skillet, heat 2 tbsp. butter over medium-low heat. Add the bell pepper and garlic and cook until tender, about 7 minutes. Add the parcooked pasta, reserved cooking water, the chickpeas, lemon zest, crushed red pepper and 1 tsp. salt. Cook until the pasta is al dente, about 5 minutes.
Remove the pan from the heat and stir in the cheese and remaining 3 tbsp. butter. Stir in the arugula and toss to wilt.