How to play the demo stage in the original Prince of Persia (1989) for DOS?

Could this be what you're looking for?
The last entry under the first header 'Download Prince of Persia 1' leads to a downloadable file, including "the first two levels".

According to the same website, popuw.com (here, specifically), the following shows a map of the "demo level":

On forum.princed.org someone seems to have remade the demo level.

You can find it here (and directly through here).
It was originally uploaded by user mk1995, and, since the file is no longer available, reuploaded by user Norbert.

So, apparently, there is a demo of the game (including two levels), as well as a demo level. I hope this is helpful.