Is it possible to get permission to fly a quadcopter around the St. Louis Arch for non-professional means
The Superintendent's Compendium for Gateway Arch National Park says


  Launching,  landing,  or  operating an  unmanned  aircraft  (UA)  from  or  on lands  and waters administered  by  the  National  Park  Service  within the  boundaries  of  the Jefferson National Expansion Memorial  is  prohibited  except  as  approved in  writing by  the  superintendent. 


Since drone flight is not listed under activities that require a permit they clearly don't approve such activity on a regular basis. The only way I imagine they would approve non-commercial activity is if there was some sort of organized event. On an individual basis I expect the answer would be no.