How can I get a copy of my Military Entrance Examination conducted at a examination station in Little Rock Ar in 1963? Records Management Center in St. Louis MO Have conducted an extensive and thorough search and were unable to locate.

You are unlikely to find the actual records of this matter, due to the time that has passed.
The best you maybe able to uncover is the results, that's assuming you are looking at a particular person service record.

Yes, I doubt that it was ever keep, except for the results and many records that are now over 50 years old did not all get keep as well as they should have. But often even parts of military records are missing pieces.

How can I get the results, If available?

If the records service does not have it, it may not exist any longer. How many years did you serve in military service back in those years?

And I guess why would one even want it ?