Does spending more time outdoors in bright sunlight (without doing exercise) improve the immune system?

Doubtful. Spending time in sunlight mostly will (1) increase your vitamin d, (2) tan your skin and (3) increase your risk of skin cancer. You may feel a little better as humans seem to flock to the sun. There is no known connection to immune system activity.
Yes. Sunlight helps our bodies convert vitamin d to its active form. Vitamin d is trending as an important immunoregulatory hormone, in addition to its vital role in calcium metabolism.