1. As the others have pointed out $2,400 peso max with 2 transactions per day general limit with a bonus $95 peso fee per transaction.

2. Manuel Tienda Leon is a bus service that leaves approx every hour is the most cost effective way to get into the city. You'll get dropped off near Retiro/Puerto Madero and then need to taxi to your hotel/apart. They have a window just outside of customs but before entering the main hall. I think there is one in the main hall as well, but I am not certain.

The most cost effective, convenient way would be the kiosk right in front of McDonalds in the main hall, Taxi Ezeiza. Going rate is about $500-$600 pesos?

Uber does work, but every time I have done it...it is a hassle. Uber vs Taxis is still an issue, the pick up points are not clear and need to be communicated by the driver. Wifi only works inside so without cell service it becomes a pain.