Cook pasta in a large pot of boiling salted water until al dente. Reserve 1/2 cup pasta water; drain pasta, and set aside.
Place pasta pot over medium heat. Add garlic and 1 tablespoon butter; cook, stirring, until garlic is fragrant, about 2 minutes.
Add arugula to pot; toss just until wilted. Add beans, pasta, and remaining 3 tablespoons butter; season with salt and pepper. Toss, adding enough reserved water to mixture to coat pasta. Garnish with walnuts.
Baby arugula doesn't last long, so buy only what you plan to use within a few days. To clean, fill a bowl with cold water, and add greens, swishing gently. Scoop out greens. Drain water off. Repeat with fresh water until no grit remains at the bottom of the bowl. Dry with a salad spinner, or pat with paper towels. Store washed greens in a resealable plastic bag lined with dry paper towels in the refrigerator up to 1 day.