For serving a display on Linux, such as when starting a Cygwin X session to your Linux server, you need to allow access to your machine. I did it like this on Fedora:
Startup Cygwin/X like this (make sure there is no Cygwin X server running):
XWin.exe -fp tcp/<servername>:7100 -query <servername>
You're probably better of running X over SSH (i.e. in Cygwin ssh -Y <server>) or installing VNC or some such application. But for this method, you do not need to install anything on the server. On a decent network, it is reasonably fast but you should not run it over the Internet, of course.