Pretty easy. You just file a claim over that one. If you already know the quadrangle and township and sector, plus some other small details, taking over the old claim is pretty much as simple as filing paperwork and paying a fee.

Is the process state-dependent? Can you only reclaim mines in public lands (and does it matter if it's NF vs BLM?), or can you claim an abandoned mine on private property?

The process is pretty much the same as mining claims are in the jurisdiction of the Federal Government. Just look up the Bureau of Land Management for your state and they should have everything you need to know on the website.