You don't need to cook the rice first. Add it to the soup and it will absorb all the flavors of the broth while it cooks, becoming flavorful and tender. However, be aware that leftover soup with rice in it will become thicker as it sets. The rice will continue to absorb the liquid even while it's refrigerated. If the soup becomes too thick for your taste, add a bit more broth to thin it when you reheat the soup.