Just a short notice on using WSUS (Windows Server Update Services) for updating a Server Core (2012 R2) installation:

in order for this to work (i.e. to be able to select automatic update option from the sconfig menu and keep it selected), Windows Update service has to be set to run automatically on that machine - this can be adjusted either by using a remote console management (mmc.exe - just add the 'Services' snap-in from the 'File' menu upon opening) or by remote PowerShell.

In addition, if you still don't get that Server Core client shown in your WSUS management, try resetting the Windows Update client / service by issuing  wuauclt /detectnow /resetauthorization.