His IT dept. probably abandoned that system, now pretty old parts, weird things happen...
I assume OP meant the monitor magically works when the system is restarted ONLY?
I wouldn't bother turning the offending monitor off then, except when system is turned off.
LCD/LED monitors use very little power.
If you must troubleshoot it, find another video card and monitor, those are the likely failure points, cable should be more than just 'oh look I won't work if the monitor off and on button is pressed'.
I vote for it being monitor, it might be requiring a warm up period before working again after turning off, I had one do that before.