Stucco textured walls are quite strong, so attaching gutter downspouts should be easy, provided you take reasonable care.
Locate the Downspout
You may already have a downspout junction in your guttering. This will determine the location of the downspout. Use a plumb line to mark a vertical chalk line against the stucco slightly to one side of the downspout junction.
Using the masonry bit, drill through the thickness of the stucco at intervals of 5 feet. Drill another hole the horizontal distance from the first hole equal to the distance between the 2 sides of the downspout straps. The second hole is just to the other side of the downspout junction.
Assemble the Downspout
Screw the downpipe straps to the stucco using the holes you have drilled. It may be easier to feed the pipe through the straps already attached to the wall, especially if you are working up a ladder. As long as the straps are tight against the stucco, there should be no problem. Just make sure you have the downspout sections the right way up—broad end at the top—and use the supplied seals.
Once the pipe is connected, you can apply stucco to the edges to help it blend in with the building.