Question: I was reading some clickbait "cleaning hacks" article, and it mentioned cleaning a mattress by using vodka in a spray bottle, then leaving it to air dry.

Is this actually effective? If so, I'd rather use something with even more alcohol, like everclear (190 proof).

Are there any other more effective ways to clean a mattress?

Answer: 
Alcohol solutions are used for removal of living tissue and oils. The ideal blend is 70% alcohol to 30% water. The alcohol damages the cell wall, and the water causes the living cell to rupture.

The best solution is prevention, or a sacrificial top to the mattress. A separate layer of high density foam at the top of the mattress is ideal. This can be aired out every so often and the cover laundered.
If you have a steam cleaner (for cleaning bathrooms/ovens etc) they are ideal for also cleaning a mattress as long as you can air it out afterwards to allow it to dry.

Simply move it over the surface of the mattress, the steam will kill any organics within the mattress and the water will cause dirt to wash away/lift out.

Once done, the best way to dry it is to leave it outside in the sunshine. The UV rays are also antibacteria.