You should be able to have the hotel get you a taxi, or get a Cabify or Uber (I prefer Cabify there) - AEP departures are generally pretty easy, except for the check-in counters, which can be crazy. Also note that your airline can get pissy with your US-sized hand luggage, if you've got the standard full-size rollerbag.

I've only departed AEP a few times and the wait for passport control has been nominal, same with the security checkpoint.

I'd try and be in a car by 3:30 am and you should be fine.

Caveat - a major holiday - cars may be in quite limited supply, and you'll want to make advance arrangements. If you want a taxi service https://www.taxipremium.com.ar/es/ is generally the most reliable.