The Rio Gallegos on the mainland has a run of browns, the Rio Santa Cruz further up the mainland has the only confirmed run of Atlantic steelhead. However, these rivers are a long, long way from Buenos Aires.
The Falklands has sea-run browns, but is expensive to get to and the fish aren't as large as Tierra Del Fuego.
The best resource for information on fly fishing in Argentina, aside from the web, is the book "Argentine Trout Fishing" by William C. Leitch, available off of Amazon.com.
The Buenos Aires area is not a trout destination, however it is very close to some world class dorado fishing. Not the saltwater fish, but the freshwater fish. Do a search on the net and you'll find some info. Best of luck!
Just as an FYI, if you plan on heading to Tierra Del Fuego, be warned that the access is generally very limited for the Rio Grande. Also, you may find the conditions to be less than perfect, with constant 30+ knot winds. On the other hand, Thomas McGuane caught a 20+ pound brown on a dry fly down there the other year...
The bottom 50 kilometers of the Rio Grande (read: the part you want to fish) is owned by the Estancia Maria Behety (sp?) and has no public access. Argentina does have a stream access law, however, from the first bridge to the town of Rio Grande is a 50 kilometer float, and you have to do it in one day. Best of luck
There are about 9 other streams on the Argentinia side of Tierra Del Fuego that have substantial enough populations of brown trout to be worth talking about, but again, access is very limited.
This isn't designed to keep you from going to Tierra Del Fuego, just to give you a heads up of what to expect. Personally, I would suggest either to fly to Cohyaique in Chile, Fue or Esquel in Argentina, rent a car for a couple of days, stop in at one of the local fly shops, and explore some rivers in that area. You'd probably have a better time, catch more fish, and not get blown all over the place. Plus, it's a heck of a lot closer to Buenos Aires.
With that being said, if you do intent to go to Tierra Del Fuego, contact me off list and I can provide you with more detailed information, as I'm heading down there in February next year. The difference is that I'll have a few months to explore, so if I can't fish one day because the wind is blowing 120 mph, it's not big deal. If your heart is set on it, however, I would contact either Kau Taupin or Maria Behety and ask if you could do a couple of days instead of a full week, be prepared it'll cost you...
If you need any more information, feel free to email me.
Best of luck and report back when you return!