In reply to mypyrex:
Have a look at the connections on the back of your new and old PC's, and the connections on the monitor. You may find that you can run a cable from the new PC using HDMI, and the old PC using VGA/DVi or whatever else you have. Then you can switch from the monitor.
My set up:
MacBook connects to monitor via DisplayPort
Raspberry Pi connects to the same monitor via HDMI
Then on the monitor I choose the input source.
The other option would be a physical device to manage the connection, something like a KVM switch (Keyboard, Video, Mouse). Feel free to post you set up here and then we can point you in the direction of the best device to suit your needs and wallet.
Also, post the software that you use the old PC for, I'm sure there are plenty of us here who can advise on it, or alternatives