They should be able to do same day pickup at the passport agency. That's a big part of what they do, last minute situations like this.

Post offices are official passport acceptance facilities. So you can apply and renew your passport there, they can provide the correct stamps and send it to agency. But they're for when you have plenty of time. They just mail it up and it takes a few weeks just as if you were to renew yourself via mail. And it can take a month just to get an appointment.