Is there a proverb that describes a self-important person of low standing?

Too big for his boots, or, alternatively, too big for his britches

Cambridge English Dictionary says

A person behaving as though he is more important than he is

This was a very common expression in the mid-20th century, but less so now. It was often used to put sassy and self-important children in their place, but was also used for adults. The Cambridge English Dictionary's example sentence referred to an adult:

He's been getting a bit too big for his boots since he got that promotion.