UBA starts in March (being in the Southern Hemisphere). You normally have to enroll in December, before Christmas. This is what I remember, please check this info. (you might be able to start the CBC in the second term, which starts in August).
For the first year you must take general courses, called the CBC (ciclo basico comun). These can be taken outside of the actual faculties in many places throughout the city and the province of Buenos Aires. Once you pass this--should take a year, it is used as a filter and some people take more time--you then enroll in the department of your choice--law, literature, medicine, etc.
When I checked a few years ago ANY university in Argentina requires an official HS diploma. The US HS diploma is NOT recognized in Argentina (Spain is, for example) so he will have to take 6 exams at a local HS to get his HS degree. He does NOT have to enroll in HS, just take the exams. I suggest a tutor for this. You need to study, but not impossible, and if you can´t pass HS you definately won´t be able to handle UBA!!!!
Some private universities don´t require this but they cannot grant you an official diploma if you don´t have your HS degree. One private film school said that they would give me a degree but on the certificate there would be a sticker saying "not legally valid". Kind of like if you go to a US university and take all the credits but don´t get the degree.
If your son wants to do this, he will need to have very strong Spanish. he should come after he graduates HS, and start studying for the HS exams while he also practices his Argentine Spanish, maybe even sitting in on some UBA classes ("oyente" very easy to do, you just need to ask the professor, no paperwork).
Another great public university is the university of cordoba in a smaller city (still 1 million people) and the people there are very nice, like to party (not in a frat school way).
What does he want to study? I think if he wants to study medicine or science, this makes a lot of sense. He will get a medical degree for basically, nothing, and Argnetine doctors are very well respected. But he needs to be aware of how the UBA works. No hand holding, but a great learning experience. Remember, the majority of students fail out, and the system is designed to weed out the weak which is very different than the US system. The UBA has a lot of prestige in part because if you can graduate it means you are motivated and can deal with adversity, a lot of the private universities (not all) treat students as customers, more like the US system.