Is there a special way to prep or finish or polish the top of an anvil that has radial patterns (looks like thumbprint lines) on it?
Use a disc grinder and start with #40 grit paper. Move on to finer grits and remember to keep it moving in a "cross hatch" patern. That will keep your surface flat. "Cross hatch" patern is useing the top 1/2 of the disc, leaning it to the left while moving to the right, leaning to the rt. while moving to the left. The patern looks like X's and keeps the surface flat.
#150 grit you can use a jitter bug sander (or a vibrating sander)
#220 grit you can use a jitter bug sander (or a vibrating sander)
You can go to higher grits depending on how polished you want it.
You can spend more time on rounding or cleaning up edges with the finer grit papers, be careful with the #40 grit it will eat up metal quick. These kinds of tools from harbour freight just are not finished off as nice as some others, but with some time spent grinding, sanding and polishing it can be real nice.