In a large bowl, whisk the oil, Parmesan cheese, lemon juice, arugula and 3/4 teaspoon of salt and 1/2 teaspoon of pepper to blend. Set the lemon sauce aside.
Bring a large pot of salted water to a boil. Add the spaghetti and cook until tender but still firm to the bite, stirring occasionally, about 8 minutes. Drain, reserving 1 cup of the cooking liquid. Add the spaghetti to the lemon sauce, and toss with the basil and lemon zest. Toss the pasta with enough reserved cooking liquid, 1/4 cup at a time, to moisten. Season the pasta with more salt and pepper to taste. Transfer to bowls and serve.