I have recently had great success cleaning a dirty finish on an organ with a Magic Eraser and Simple Green, then waxing w/furniture paste wax. The finish was shellac/french polish, which is likely the same on an old piano.
Do not let the moisture sit for too long, and do not use any sopping wet things on it. My organ case was solid wood, your piano is certainly veneer. Water will loosen it with not good results.
If the finish is really bad (alligatored) then a fine sanding by hand with 240 paper, and several new coats of shellac would bring up a nice finish without removing all the original color, as stripping would do. You may then consider a wipe-on poly varnish for the top coat so as to add more water and alcohol resistance, the achilles heel of shellac on a tabletop.
I assume you mean the lower case panel over the "trapwork". That may be interesting if you find a good way to finish off the edges. Consider adding a skirting/apron around the bottom, because the wood frame is not really heavy enough as is to be much good as a table.
Casey

Can you post a picture?

If it's alligatored old finish a 50/50 mix of shellac thinner and lacquer thinner will dissolve the old wax and varnish build up without affecting the wood very much.

Apply it with medium steel wool - spread it on a small area - let it sit a short time and then GENTLY scrub off the old wax and wipe off the gunk. Repeat until it's clean, with the final cleaning down with 0000 steel wool, then apply the finish of your choice.

Here's that cleaning method with an extremely cruddy, alligatored old dresser from a swap meet. It had decades of the classic beeswax and linseed oil polish, topped off with commercial products and hand grime.