Is it permissible to fly a small quadcopter (<1 kg) around St Louis Arch for non-commercial means? If so what permit/permission is required?
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.