Bring a pot of water to a boil; add a generous amount of salt. Cook pasta until al dente according to package instructions. Reserve 1/2 cup pasta water; drain pasta.
Add 1 tablespoon butter and the garlic to the pot; cook over medium heat, stirring, until garlic is fragrant, about 2 minutes. Add arugula; toss just until wilted.
Add beans, pasta, and remaining 3 tablespoons butter; season with salt and pepper. Heat, tossing, until butter is melted and beans and pasta are warmed through, about 1 minute. Add enough reserved pasta water to create a thin sauce to coat pasta.
To serve, divide among shallow bowls, and garnish with walnuts.