Put the potatoes in a large saucepan, cover with cold water and season with salt. Bring to a boil, then reduce the heat to a gentle boil and cook until fork-tender, 8 to 10 minutes. Remove the potatoes with a slotted spoon to a large bowl, leaving the water in the pot.
Fill a separate large bowl with ice water. Return the pot of water to a boil, add the green beans and cook until crisp-tender, 5 to 7 minutes. Drain and transfer to the ice water. Drain the beans again and add to the potatoes. Add the cherry tomatoes.
Combine 2 cups arugula, 1/4 cup olive oil, the pine nuts, Parmesan, half of the garlic, half of the lemon juice, 1/4 teaspoon salt and a few grinds of pepper in a food processor. Process until smooth. Pour the pesto over the vegetables and toss to coat.
Heat 1 tablespoon olive oil in a large skillet over medium-high heat; add the red pepper flakes and remaining garlic and cook 30 seconds. Add the shrimp, sprinkle with the lemon zest and season with salt and pepper. Cook, turning once, until pink, 3 to 4 minutes. Remove from the heat.
Toss the remaining arugula, 1 tablespoon olive oil and the lemon juice in a bowl; season with salt and pepper. Divide the arugula among shallow bowls and top with the vegetables and shrimp.