If you need to run it after the X server is started, it's probably more appropriate to start it using your desktop environment's settings, rather than attempting to cobble up something with systemd (although you could launch it with systemd in user session mode, but including your X launch in there may take some more work that would only be tangential to your goal, depending on how complex your setup is).
If you're using
xinit, just add the program to
Otherwise, find your startup settings in your desktop environment, and add the program there, so that it launches shortly afterwards.
Since it is now clear that you are running GNOME 3, you can run
gnome-session-properties from Alt+F2, and add the program to the startup list there.
As the program needs superuser permissions to function, you need some way of elevating without being prompted for a password. To do this without blocking at the prompt, you can make your user able to run it with
/etc/sudoers. Edit the file with
sudo visudo, and add the following line at the bottom, edited as appropriate:
youruser ALL=NOPASSWD: /path/to/program
You can then run the program using
sudo without being prompted for a password.