Downunder - Retiro refers to this strange little place down by the Retiro bus station that you might have to go to collect a parcel sent to you internationally if it's anything bigger than an envelope. You'll (eventually) get a card telling you it's waiting for you there. You might not get the first or second notification, so it will you'll have to pay a fine for not collecting it sooner even though you've gone straight there as soon as you got the card. You can't collect it without the card. Then you'll have to wait whilst they shout out a list of numbers over a crackly tannoy, one of which will be yours (about an hour after you've arrived if you're lucky) and have to go through a door to collect your parcel, where you'll confirm the contents (they might even open it these days) and be on your way.
Ezeiza - similar process but you'll have to go to a place near the airport (I never had the pleasure as after my first trip to Retiro, I told everyone not to send me anything). You'll probably have to pay a random tax on your package and the contents will be checked.
The other thing to expect is that it might arrive in a few weeks, but more likely you'll get it 4-5 months later if at all. The parcel I was collecting was a book from Amazon that I needed for my work (impossible to get in Argentina). It took about 4 months for me to get it. Around the same time we had a couple of packages sent by family, neither of them containing anything valuable (some kids drawings, birthday cards, a pair of slippers!), neither of them ever arrived. This was a few years ago before they tightened up on imports, which gave the aduana (import tax people) even more of an excuse to hold on to your stuff.
So if you have an option of collecting yourself, then I'd say that this is the fastest and least stressful way of obtaining the ring! You might be lucky as JoshF was, but if there's another way, I'd go for that over using the post.
I don't live in Argentina any more, and to be honest I'd say the difficulty getting mail is in the top 10 (if not 5) of my reasons for this. Seems a very petty thing, but when you're far from your family, it's a real comfort to get letters, drawings from nieces and nephews etc. Hopefully this is something that will start to change with the new government.