Bring 7 cups of water to a boil. Add fettuccine, stirring to separate the noodles. Cook until just tender, 7 to 9 minutes. Reserve 1/2 cup of the cooking water and drain.
Meanwhile, heat oil in a large nonstick skillet over medium-high heat. Add shrimp and cook, stirring occasionally, until pink and curled, 2 to 3 minutes. Transfer the shrimp to a bowl.
Add butter to the pan and reduce heat to medium. Add garlic and crushed red pepper; cook, stirring often, until the garlic is fragrant, about 1 minute. Add arugula and cook, stirring, until wilted, about 1 minute. Reduce heat to low. Add the fettuccine, yogurt, lemon zest and the reserved cooking water, 1/4 cup at a time, tossing well, until the fettuccine is fully coated and creamy. Add the shrimp, lemon juice and salt, tossing to coat the fettuccine. Remove from the heat and toss with Parmesan.
Serve the fettuccine topped with basil and more Parmesan, if desired.