My makeshift headset press consists of two stiff, flat plates of metal with a hole in each. One goes on the top, one on the bottom. I then put a wheel quick release all the way through the metal plates and head tube, screw on the quick release nut and then operate the quick release just like when using it to clamp a wheel. Then release the quick release, tighten the nut by one or two turns, and operate the quick release again. This slowly presses the cups in quite nicely. It is better if you move the metal plates and quick release rod around a little bit each time you tighten the nut a bit more so that each side of the cups gets pressed in evenly. Obviously, it only works when the head tube plus plates is shorter than you quick release, but it should be OK for head tubes up to about a 140mm head when using a MTB rear quick release. I sometimes need to use some spacers below the nut if I run out of thread on the quick release to tighten the nut more.

To remove old headsets, an old screwdriver and hammer can get the job done if used carefully, but is not ideal, so I now have the correct tool that was described above - like a metal pipe with the ends splayed out.