Preheat a grill to medium. Lay out four 20-inch sheets of heavy-duty foil on a work surface. Bring 6 cups salted water to a boil in a large deep skillet.
Add the linguine to the boiling water and cook, stirring occasionally, until pliable but not cooked through, about 5 minutes. Transfer to a large bowl using tongs, add 1 tablespoon olive oil and toss well. Add 1/2 cup of the cooking water and toss well (reserve the remaining cooking water). Add the shrimp, garlic, lemon zest, oregano, red pepper flakes and the remaining 1 tablespoon olive oil; season with salt and toss.
Divide the pasta mixture among the foil sheets. Drizzle evenly with the wine and 1/2 cup more pasta cooking water total. Top each with 1 tablespoon butter. Bring the two short ends of the foil together and fold twice; fold in the sides to seal, leaving a little room for steam to circulate.
Grill the packets seam-side up until puffed and the shrimp and pasta are fully cooked, 10 to 12 minutes.
Carefully open the packets and add the arugula; toss to wilt slightly. Sprinkle with the parmesan and serve with lemon wedges.
You need to parboil pasta before you put it in a foil packet. Boil it about half the time recommended on the label.