Server Core can be configured via group policy to utilize your Windows Server Update Service (WSUS) environment. The problem is this. The server core will reboot. Unlike the GUI versions of Windows, you cannot receive little popup windows asking you to reboot the computer to complete the update installation. Server Core will reboot. To get around this will require you to manually update the server. To do this, you will need to download the .MSU files for the patches and install them using Windows Update Stand-alone Installed (Wusa.exe). In particular, you will want to add the /quite and the /norestart switches to the command line. This will prevent Server Core from reboot until you have installed all your updates and perform the reboot manually. Instructions for this procedure are in the link below.