I would make sure the soil the plant is in is not overly wet for starters. If the soil is still wet in the bottom, wait for it to dry out for a couple of weeks before giving any more water.
If you water it "from above", you could be getting water on the leaves, where fungus will take hold and cause your black spots.
Unless the spots grow, I would not cut the leaves off as you would be cutting a large part of the plant off for something that is most likely not systemic(throughout the plant).
I would "wait and see", let it dry out, if it is in full sun on a windowsill, move it to a bit more shade and away from the scorching sun as reflected by the glass, if it's in a shady spot, move into a little more sun.
