<< am familiar with the theory of relativity, and time displacement(not that I understand how it works) but my question was, Will I catch the ball?(If going the speed of light?>>
I think you have set-up an argument that has a problem in its logic, so any answer will not have Truth in it.
If the argument goes, "will the light beam that goes around the world from the top of the building, shine on the ball when it reaches the ground" ; I would say yes.
You as an entity of inertia/momentum cannot do the experiment, practically nor in thought.