Hi Sam!
I did it last year! So I can give you some pretty useful information about UBA.
The first piece of information that I think is the main punto de partida for you to understand how to go about enrolling yourself is that UBA is not governed like a US university. This is something a newly hired professor with his PhD from UBA in Anthropology at my university shared with me, and it helped me understand the system much better. EACH FACULTAD, or college, governs itself and is 100% independent in governance structure. It is unlike a US university or a private one in which there is an overarching university administration that sets standard policies for the entire institution. So after studying in UBA, to talk about THE UBA does not really make sense to me...there really is no one UBA, but a bunch of UBAs, each with their own policies and culture. This means policies about entering foreign students will vary from facultad to facultad.
I studied at the Facultad de Ciencias Economicas in some senior level courses in economics and also in the language center CUI. I tell ya, in FCE, they do not throw you bone just because you are not a native speaker of Spanish! I worked my butt off to pass, and in all my classes except one, I think the other couple foreign students -all from Europe - all did not pass the course, though they seemed to have a social life outside of school that probably had a lot to do with it 'cause you HAVE to read most of the material to pass....though, flunking is not a "shameful" thing in la UBA, like we think of it in the US (oh no! you flunked), as half the students often are not passed in some of the hardest courses. They have more of an attitude of, "Come back next time and try again to raise the level of your understanding," which is a great attitude. Professors could care less if you turn in your homework or show up; all the effort and responsibility is in the hands of the student. They don't spoon food you ANYTHING - which can be frustrating - they blurt that a new article is at the photocopy store, which you just barely catch at the end of the two hour lecture and confirm with your compañeros just to be sure...then, it turns out you never get to the article and don't need it or it's 1 of 4 parcial questions. I have so much love for FCE: people come into the middle of classes and tell their hardship story just like on the subte and ask for coins, and a pounding drowningly loud march could be going on, and the teacher just keeps lecturing. It's so bad ass. And if you love living in BsAs as you do, you will love this kind of chaos and challenge!
That said, in order to get in FCE asked for a transcript, a letter from a language school or authority affirming I already had dominio over the Spanish language, a letter of recommendation from a university official, proof of medical insurance and of course, letter of motivation from me, and curriculum vitae. FCE has an "International Student" office, which has what seems to be one staff person, maybe two. I just emailed anyone I could find online for Estudiantes Visitantes and asked politely who to direct me to for enrollment; since you are here, you could just go directly to the centro de estudiantes. If you are doing this for the visa, ASK specifically for the office or people who deal with enrolling international students because they will be the ones to get you the letter to take to the Immigration Office. They wanted all documents translated, which I got away with doing myself and not saying anything.
In order to have all the documents Argentina requests online for a student visa (e.g. apostilled criminal records and birth certificate, etc), you need to be applying and admitted to a year long course of studies for the year long student visa. Otherwise, for a semester, I only needed antecedentes policiales from Argentina (my criminal record for a country at the time, I had lived in a couple weeks).
FCE was free as I was a visiting undergraduate student, but as I always say, you pay in sweat equity, but it was well worth it. Sometimes the sweat is not productive sweat-because you can spend 3 sleepless days on a TP and get a three and find out you did it all wrong and not know why. It is not free for graduate studies, but as I remember, it would be about $2000 USD for an entire Masters there, though again you pay in other ways there. The solidarity and the seriousness of the students was a great change from my US university, and the politics inside is even more fascinating (that could be a whole other post)!
CUI, or the languages institute, is 180 degrees from Facultad de Ciencias Economicas in its level of demand and the stress you will experience taking a course....as I am sure it is from the other facultades. It charges for its Español para Extranjeros, as does the Language Laboratory under the FFL. CUI is technically under the Facultad de Agronomía, but I heard better informal reviews from expats about CUI than the Spanish classes offered at the Language Laboratory under UBA's Facultad de Filosofía y Letras
http://www.idiomas.filo.uba.ar/extranjeros/ingles/extranjeros.htm. I took the month long Ciclo 7 course and had a class of 4-5 student (depending on who showed). It was in no way a DIFFICULT class, but I got a lot out of it. I fell in love with one of my teachers, as she was the first Argentine woman I met with her intense, beautiful way of speaking and her great knowledge and fascination of culture and literature and ideas. It was the perfect intro to Argentine Spanish if you already have a strong base of book Spanish, but I would not take it if you have lived here a while with a strong base of Spanish because the value I got out of it was more as an indirect intro to Argentine vocab and mind.
Hope this helps!