The date written on your ARC is for reference purposes only. Your ARC expired when the school sent notice of the end of your purpose of being here. That means you could be looking at an overstay of months.
No, the school is in no way obligated to inform you. The ARC is your responsibility. It is very easy to get an extension just by showing an airline ticket or any school documents. Yes, overstay is a biggie because it is very avoidable.
Since all is done, you must go ASAP to NIA. If you show contrite, they will help out. They are very willing to help if you collaborate. I wouldn’t risk just hoping to pay the fine at the airport. It may end up in a lost flight when they turn you down and make you go back to Taipei to pay, as it has happened to others.
What Icon said. Go visit the NIA office on Monday morning with your ARC card, your passport, your school ID, and a copy of your air ticket. Put on your most concerned face and ask if the situation will be alright.