How common is it for a man to be shorter than his partner

Very likely

A study finds that in the U.K. 92.5% of couples the man is taller than the woman. 3.4% of couples were the same height, and 4.1% of couples the woman was taller.

The United States is similar, with 92.2% of men being taller than their spouses.

Of course, men tend to be taller than women anyway, and much research has been done on the correlation between height and IQ (1,2,3)

While the studies I mentioned were of real life couples, another study shows people also say height matters - mostly on the woman's side. 49% of women reported that they preferred a taller man. 13.5% of men reported that they preferred a shorter woman. This isn't the only way women might be more particular than men when choosing a mate. Some evidence from OK Cupid shows women give much lower ratings of men on a scale from 1-10 than men give women.

Some ways that men are more particular than women might be that they consistently find 20 year old women at peak attractiveness. Meanwhile women consistently prefer men a few years older than them until about 30, and then a few years younger.

This study shows men of average height have the highest reproductive success. Some others (1,2) show that being taller than average is optimal for reproductive success.