Cook the pasta. Bring a large pot of water to a boil. Salt generously. Cook bucatini pasta, stirring occasionally, until al dente. Reserve about ½ cup pasta cooking water before draining.
Prepare the sauce. While the water comes to a boil and pasta cooks, begin the sauce.
Heat 2 tablespoons olive oil in a large skillet over medium-high heat until it shimmers. Add the tomatoes. Cook until soft, shaking pan occasionally, about 5 minutes. The tomatoes should pop and brown a bit during this step.
Add remaining oil. Heat for a minute or two. Add garlic. Cook. Stirring occasionally until garlic is golden brown.
Add tuna and break up the tuna with a spoon. Stir. Add the olives, followed by the spinach. Cook until spinach is wilted. Reduce heat to low. Add the lemon juice. Stir.
Toss the pasta with the sauce. Toss the drained pasta with the sauce. Season with salt and pepper to taste. Adjust the thickness of sauce with a splash of pasta cooking water if needed.