the headset press just installs the cups, correct. You can make your own press from some 1/2? thick threaded rod, and nuts, and the largest washers you can find (larger than your Outside Dia of your cups). total cost about $5-$7. Not nearly as easy to use as the park one, but much cheaper. if a LBS will let you use theirs then do that, using their tools is even cheaper, and the Park tool maintains perfect flatness of the cups while pressing them in better than the home-made job I described. If you build your own, sandwich 2 washers between 2 nuts as your base, and then press 2 washers down with another nut to compress the headset. This will at least keep 1 of the washers completely square and less prone to wobble.

You'll probably be replacing the race on the fork too, as they don't all match. the old one can be removed with a hammer/screwdriver tapping lightly from the bottom, careful not to chip paint off the fork. the new one can be tapped into place with a piece of pvc pipe of the correct diameter, which should dent before the race, since it's softer.

The new cups and race should be fully seated before compressing the headset with bearings..etc. Don't use your headset to compress the cups into place, it can cause divits in the cups and races where the bearings were, and you could even strip threads.