Cook the pasta according to the package directions, reserving ½ cup of the cooking water.
Meanwhile, heat the oil in a large skillet over medium. Add the soppressata and cook, stirring occasionally, until crisp, 5 minutes; remove with a slotted spoon to a paper towel–lined plate. Reduce heat to medium-low. Add the garlic and red pepper; cook, stirring occasionally, until the garlic is soft, about 2 minutes. Remove from heat. Stir in the sour cream, lemon juice, black pepper, ⅓ cup of the reserved cooking water, and the salt. Add the pasta and toss to combine.
Transfer the pasta to a large bowl; add the arugula. Toss until the arugula wilts, adding more cooking water as needed. Top with the soppressata.