How to protect an old, internal windowsill?
I generally use a clear-coat Varnish finish on my outdoor painting projects to reduce wear damage and to extend the life of the paint. You can get this as a spray or a paint. This is the easiest and cheapest solution but provides only moderate protection.
For superior protection:
You could attach a piece of plexiglass or tempered glass with adhesive or standoff mountings. I'm not sure if the adhesive would be visible between the glass and the voids in the surface. Using an attractive glass with standoff mountings would provide a very clean and modern look. A quick google search for glass standoffs shows that you can get them as short as .5in/1.25cm.
A final thought would be to make a mold around the windowsill and pour in an acrylic resin. This would fill in the voids in the surface and provide a very sturdy glass-like finish.
Here is a link to pouring acrylic on a table. You can adapt the instructions for your windowsill.
Note that using an adhesive or acrylic resin is more a less a PERMANENT installation. It will be very hard to remove either. The varnish or glass+standoffs is considerably easier to remove later if you want to repaint or change the look.