First of all, I doubt you are talking about visa on arrival. Visa on arrival is where you apply for and pay for a visa at the airport and you get a 15-day stamp for Thailand. You are most likely talking about visa exempt, which is not a visa at all, it's just a free entry stamp for 60 days.
They have been cracking down severely on people who abuse visa exempt and tourist entries in general. Each immigration officer has full discretion on who they will and won't allow into Thailand. Some seem to use a general 180 day cutoff as to when it seems like you are trying to live in Thailand without a proper visa. Also anyone who has an ED visa at all in their passport is likely to face extra scrutiny and have very high odds of being pulled out of line and questioned about their ED visa, why they are coming to Thailand now, etc. It would be HIGHLY advisable not to fly into Thailand if you are just finishing an ED visa, as this is the most abused visa type currently and is just asking for trouble. Use a land border bounce agency to drive you to a land border and facilitate you leaving and returning without any issues, but only AFTER you have already cancelled your education extension. Some land borders will not let you exit if you have an un-cancelled ED extension and no re-entry permit.
Or if you want to do it yourself, go to Udon Thani and cross into Laos at Nong Khai and then return there. Once again, cancel your ED extension first as you are required to do this. But that land border is known as one of the easiest in Thailand.
it doesn't sound like you have enough time to get a new visa at this point if your flight is next week. It's already the weekend so even if you file a new visa application it won't even be looked at until Monday.
Just forget about it. If you have an Italian passport you'll receive a 60 day visa exempt stamp if you don't have a visa. That's the same amount of days as if you did have a tourist visa.
No way to say if it will affect anything in the future, as all of those decisions are made at the time by the immigration officer dealing with it. It's certainly not ideal to have an overstay stamp in your passport, but it shouldn't affect your current stay other than the fine.