@Jakehotep's answer is rather scattershot. @mpez0's comment/answer is a pain unless you're a dedicated X-windows user from way back.
Just turn on screen sharing on the old computer.
System Preferences, Sharing, Screen Sharing. No need to turn on "all the sharing prefs" to share the screen, which is the current-era terminology for (exactly) what you want to do.
The old computer should show up under "shared" in the left-hand side of a Finder window on the new computer. Connect from the new computer to the old computer, click the button in the upper right for "Share Screen", log in, you're there. The one non-intuitive thing I've found is that you need to log out of the old computer before closing the connection, unless you are happy with the old computer sitting there logged in (at its local screen.) And of, course, anyone at the location of the old computer can watch what you do on the screen.
On the other hand, you can just remove the monitor from the old computer - the main thing I use screen sharing for, having gotten it turned on during one of the times I had the local screen working briefly, is to get some use out of a MacBookPro which had the defective Nvidia graphics replaced with the same defective Nvidia graphics, which of course failed again, but Apple would not replace again. Screen sharing into it works, even though it can't display anything on the internal, or an external screen.