English equivalent to the Japanese saying "高転びに転ぶ" - A haughty man should tumble down?

The Biblical phrase, "Pride comes before the Fall"

Proverbs 16:18 http://biblehub.com/proverbs/16-18.htm

The biblical proverb (Proverbs 16:18)

"Pride goeth before destruction, and an haughty spirit before a fall," has been rephrased as the frequently used proverb "Pride goeth before a fall" or "Pride goes before a fall."

The most common proverb about pride is certainly the one quoted in the other answers, Pride comes before a fall.

Not quite about pride, but about the less attractive side of ruling is a saying,

Uneasy lies the head that wears the crown.

Though stated in those words by Shakespeare (Henry IV. Part II, 1597), it is a very old saying from before the time of the Greeks, and is told most famously in the story of Damocles and Dionysius.

Regarding pride and position, there is the saying

The higher you climb, the greater the fall.

That may have its roots in the Biblical account of the tower of Babel, but is also seen in the story of Icarus. There is no hint here, though, of rulers.

Here are three sayings about proud people getting their comeuppance.

Saying "the bigger they are, the harder they fall"

said to emphasize that the more important or powerful a person is, the more difficult it is for them when they lose their power or importance. - English Dictionary (Cambridge)

Saying "...is riding for a fall"

(informal) Be acting in a reckless way that is likely to end in trouble or disaster: "with your present attitude, you’re riding for a fall." oxforddictionaries

Saying "Too big for his boots"

too big for your boots UK (US too big for your britches) informal › behaving as if you are more important than you really are: He's been getting a bit too big for his boots since he got that promotion. (Cambridge)