Linux usually works perfectly fine with old hardware, a Intel Core2 Duo should be completely enough. That's even 64 bit hardware. Worst case you have to swap out Gnome with something more minimal, but that's unlikely.
I'm currently running Manjaro with i3wm on a laptop with a core 2 duo. It came with a couple of simple conky widgets enabled by default and those probably took 5-10 seconds to load. Just boxes, one with static text and the other with load statistics. Resizing windows is a little painful, as is opening them.
I3 runs reasonably well, but Gnome would probably be too heavy to make for an enjoyable experience.
I3 runs reasonably well, but Gnome would probably be too heavy to make for an enjoyable experience.