Is there are large building transformer (welding machine, Xray machine,
large motor machine) on the other side of the wall? All of these can
give a monitor the shakes; ditto fluorescent desk light and pencil
sharpeners. A UPS is unlikely to help since most of the time on most
UPS's they just pass the wall current through without modifying it in
any way.
The shakes can come through the air as magnetic changes or through the
power wireing. If you rotate the monitor 90 degrees does the problem
diminish (or get worse)? Then the problem is through the air. When you
moved the monitor 10 meters did you use a different wall socket? Try
using other sockets in the room; you might find one that is a lot better
(on the other side of the neutral.)
I believe if the problem is through the air, an LCD monitor would be
immune. If it is through the power wiring, another (better) CRT might
fix the problem. I don't know whether an LCD monitor would
automatically cure things.