Hello I am wondering if it’s possible just to buy a ticket and pay a fine at the airport?

I have heard a case that someone only overstayed for no more than 24 hours and the China immigration gave her an administrative penalty, and she paid the fine at the airport and get on the plane. But you overstay for one month, I'm afraid you need to go to the local PSB to state your situation, pay the fine, and apply for an exit at the exit and entry administration and leave China asap.