Just shut down both machine, remove the memory (RAM) from both machines and just swap them over.
The memory does not hold any of the OS, that is stored on the Hard Drive and is not involved in the changing out of the memory. Your OS will be there when you start up the machine again, assuming you do no damage when changing out the memory.
A word of caution. The memory is in the base of this iMac and to access it you need to remove the bottom cover.
To access the bottom you will need to lay the iMac down. I found that a pillow will provide the necessary support for the screen and allow the base to stand vertically.
Have you changed out this type of memory before? It is basically Laptop type memory and not like you see in a full size Mac or PC.
I have two G4 desktops. The one with the swing arm. One has 256MB of memory and the other has 512MB of memory. I would like to remove the 512 and install it in the G4 that currently has the 256. I am not a huge computer guy so my question is if I do this do I lose my OS. The G4 that I want to install the 512 in has OSX 10.4.11 The other has OSX 10.3.9. These computers don't have a lot that can't be replaced. I can reinstall ilife and office. My main concern is that I don't change the OS. I'm not sure if the OS is on the HD of Memory. Basically when finished I want the G4 that I install the 512 to have OSX 10.4.11. Would be thankful if someone could help me out here.