It's android based. RetroArch is usually what you want on a shield nowadays if you want to do them for free. You can find which are the best cores to run elsewhere.
I don't really know the power of a shield, but apparently it runs Beetle PSX just fine (which is something a whole lot of devices struggle with, meaning it'll run just about everything you care about when it comes to hacks). That said, Beetle PSX is not the best in the world for running hacks easily for various reasons. Look into ePSXe for Android if you have any trouble on that platform.
EDIT: Oh, and there are patchers on the store. You usually want either ROM Patcher or Unipatcher depending on what you need it for (the former's much better with patching multiple times due to being able to not create backups, the latter can deal with things like SNES headers).