Are there any ways to clean a mirror without using water or any cleaning spray?
You can use a mechanical cleaning method
You don't need any liquids at all.
If the mirror is made with float-glass which is quite flat and free from ripples, you can get excellent results with a razor blade at a sharp (heh) angle to mechanically 'scrape' the mirror clean. Use the same angle as you would to 'hone' the blade but push instead of pull the blade. This technique can cleanly remove finger prints and other surface dross. If you're careful, you can clean the mirror surface with a single blade before it gets too dull.