Is it possible to install the WSUS role on a Windows 2008 R2 Server Core installation? Alternatively, is it possible to install this role on a Windows 2012 Server Core?
Not really. Run
That said, it's mostly possible (but I sure as hell wouldn't do it). Here's a link to a guide on how to force WSUS onto a Server Core 2008 R2 install, but again, I certainly advise against it, and seeing as how you'll have to install a SQL instance on another machine (since MS SQL Server and their internal database require a GUI), the value of doing so seems rather questionable.