In large pot of boiling, salted water, cook the pasta until al dente. Drain, reserving 2 cups cooking water. Return the pasta to the pot to keep warm.
Meanwhile, in a large skillet, heat 2 tablespoons olive oil over medium heat. Add the bread cubes and cook, tossing occasionally, until golden and crusty, about 3 minutes. Season lightly with salt and pepper. Transfer to a plate and set aside.
In the same skillet, heat the remaining 2 tablespoons olive oil over medium-high heat. Add the garlic and cook, stirring, until golden, 1 to 2 minutes. Stir in the arugula, then pour in 1 1/2 cups of the reserved pasta cooking water. Cook, stirring occasionally, until the greens are wilted, about 3 minutes.
Add the pasta, tuna, lemon juice and remaining 1/2 cup cooking water and toss until heated through. Add the cheese, season to taste with salt and pepper and toss again. Top each serving with the reserved croutons.