"As a former beat cop, Ron Huberman, the new chief of public schools in Chicago, learned long ago that violence among young urban people could not be solved simply by hauling ever larger numbers of children off to jail."

what does beat mean in this context? a exhausted, puzzled cop? how can the new chief be tired already?

1. tired out: completely exhausted ( slang )
2. puzzled: unable to understand or think how to proceed ( informal )
#It has me beat.
3. of Beat Generation: relating to or produced by members of the Beat Generation

It's a special meaning of the word 'beat'.

Years ago, a uniformed police officer walked around the same streets every day. This was referred to as walking his beat. Hence, a beat cop refers to an ordinary cop.

Today, police officers usually ride around in cars where I live, so the term 'beat' is not often used in a police context.

There's also a quite common expression:

"on the beat"

This is an example I found on COCA:

I think it's time for us not to be the policemen on the beat in the city of Baghdad