There is no reason why the old (very old) 80 Gb hard-drive should not fit in the new PC as hard-drive are pretty much a standard size for a desktop: 3.5" and 2.5"for a laptop. We now have 2.5"SSD hard-drives but I wwould worry about them.
So I suggest getting an external hard-drive caddy which will take IDE and Sata. Something like this. Caddy. It will be just a case of undoing a couple of screws sliding your HDD in connecting power and data cable to it, putting the face plate back on then plugging it into your PC with the supplied USB cable.It should be automatically seen by Windows and hopefully you will be good to go.