How do you run Drupal core-patches and is there a single, general way?
Drush make is a method to maintain patched releases of contrib and core. It is the only organized, non-custom, method I know of for patch management. The makefile is self-documenting. By "self-documenting", I mean that the makefile contains a list of modules, their versions, and URLs of applied patches.

Create a makefile of an existing Drupal site with:

drush generate-makefile mysite.make


Note, as far as I know drush generate-makefile will not detect applied patches. The example makefile demonstrates how to reference patches in your makefile.