Western Union is the way to go.
I suggest you begin at the webpage
https://www.westernunion.com/us/en/home.html and initiate the transaction from there.
Just don't try to go to a small branch office. Visit the main downtown office at Cordoba and Montevideo.Try to get there before 10:30am on a weekday, and avoid the first three business days of the month. They will give you something very close to the Blue rate.
To pick up the cash you'll need your passport. Also you will be asked for the transaction number, and the amount in pesos; these will be emailed to you, in addition to being given to you when your transaction is confirmed. WU will probably call you the first time or two you use their service.
In addition to paying you slightly less than the Blue rate, they will charge you a fee of 10 USD for the first hundred dollars, and 5 USD for each hundred after that. So $15 fee for 200 bucks, $20 fee for 300 bucks, etc.
Buy and wear a money belt. Put most of your cash away before you leave the window.
Off topic but related-
I would advise you not to bring an expensive new iPhone to Argentina. If you must do so, don't take it out on the street; find a coffee shop to enter and use your phone there. Don't wear an expensive watch or jewelry. Be cautious, but not paranoid; this is Buenos Aires, not Juarez. You will be fine.
If you want a nice hotel downtown for a reasonable price, check out Uno Suites at 1944 Rivadavia. You can get a King Size bed if you ask for it. There's a gym on the top floor, and a sun lounge. Powerful air conditioners, key carded rooms, and nice bathrooms. Just don't drink the free coffee; it's horrible.
www.unosuites.com