I broke a nice tea cup by mistake.

The phrase "by mistake" means accidentally. We do not say "on mistake" or "at mistake."

I promise I didn't scare the family dog on purpose!

The phrase "on purpose" means intentionally or deliberately. It is the opposite of "by mistake."

At least I arrived for my haunting shift exactly on time.

"On time" means happening at the planned or correct time, without being late.