How exciting to have your great-grandmother's rose! The black spots look like Black Spot, a common rose fungus that isn't fatal. It's usually caused by excess moisture, either moisture in the air or excess precipitation. Most of my roses display signs of Black Spot after the spring rains. Typically, those leaves will fall off the rose and will be replaced by nice new leaves.
Other than that, the only problem I see is the need for the rose to compete with grass and other vegetation for moisture and nutrients. I would pull out anything that's growing too close to the rose.
