Sunday nights have bad traffic but usually on the opposite direction, i.e. lots of people coming in from (their weekend homes in) the province towards the city. Still, I would allow a bit more time than you are considering and probably would leave Palermo at around 8.30/9pm. Worst case scenario you spend some more time at the lounge!

I'll be flying out of EZE at 7.00 am weekday (Monday).
What time should leave San Telmo? 4-4.30 am?
Do you think uber might be available at those ungodly hours or rather should I stick to a taxi?

You should be at EZE in 25 minutes. I don't know what airline or class of service you're flying (re: check-in line), but your entire security and migraciones experience isn't likely to be more than 15 minutes. I would leave at 4 just to be on the very safe side. I would also make a Premium Taxi http://taxipremium.com/ reservation. Make it on the phone. I once made one online and they didn't come. (This was some time ago.)

Thanks a lot. I was also under impression that my night will be ruined :-)
I'll be flying GOL to Rio and then continuing to Europe.

BTW, how do you handle the phone cost situation in Argentina?
Using my European SIM card means I'll be in the "roaming mode" which is pricey as hell.
Do you buy a local prepaid SIM card to make local calls? If so, are they available at AEP/EZE airports? And what tel company?

You can buy a local SIM card in the cuty centre . easily available in kiosks and locutorios . Not sure if they are available at the airport though.

Mobile service varies acccording to where you are in the city . I use Personal which works for me . Movistar and Claro are the other options