Plan so you can go and pay a day before and stay at a hotel for the night and then take your flight the following day.
Aeroparque Jorge Newbery, (07:00 a 22:00 horas). First you go to immigration office at the airport. Hand your passport and tell them you need to pay the overstay fee. They give you a printout and you take it to the bank and pay. Then bank gives you a receipt. You take it back to the immigration office. Then they give you more papers. You give one to the airline at the check in. You give another one to the agent with your passport at the passport control. Don't wait for him or her to ask for the paper.
