Well, you are definetely very lucky to be heading there. At the same time, I will tell you that it will be hard. Remember you will be making pesos, so it´ll be ok for you living there (as long as you budget yourself… I´m teaching English in Madrid now and it´s a lot easier here). It might be hard if you want to travel outside the country, even though Brazil is a possibility as well as the beaches of Uruguay. Traveling within the country is awesome too, there is way too much to see. And cheap. Consider visits to Patagonia, Mendoza, Bariloche, Cordoba, and of course Iguazu Falls.
As for Buenos Aires, I highly recommend you stick to the following neighborhoods: Palermo, Recoleta, Belgrano, San Nicolas, Montserrat, San Telmo, Caballito, Almagro and Balvanera. The rest is too far to ever consider… at least for me! So the first three might be a little more pricey than the rest. You could expect to pay maybe like 600 pesos a month. Normally a rent is like 400 pesos a month. First of all, don´t let them screw you over bc you are a foreigner. They will always try to do that, charging you dollars instead of pesos. My advice, be very weary of people in Argentina, at least until you are acculturated. In our culture, outsmarting others and screwing them over runs rampant, sadly. Besides our huge egos, arrogance, etc. It´s still wonderful though!! :D In any case, I recommed you stay at a hostel (I enjoyed one on Belgrano and Venezuela. If you go down Belgrano heading to the Rio de la Plata (downhill), make a left on Venezuela and it´s at your left side. You won´t see it, it´s like hidden. Ask around and they´ll tell you. I forgot the name, but it´s something like 4 dollars a night. Maybe a bit more now, but it shouldn´t be more than 12 dollars. Anyways, check www.studentuniverse.com for hostels there, and chose what you would like. First off in Buenos Aires, get a “Guia T”, it´s a guide of the city, with detailed maps, metro routes and even bus routes. It´s like 2 pesos and very conspicuous so you don´t have to look like a tourist. Still, most locals use it anyways.
Take your time looking for an apartment. Or share one with people you trust. I cannot insist on how hard it will be for you to get something cause they will constantly try to screw you over. Try to make friends with locals that also are looking for apartments and get them to call and do the appointments, etc. You can also try www.craigslist.com to start looking withing the english speaking community. So, for a cheap apartment expect 400 pesos. Let´s say 500 with bills. That´s like 150 dollars. I know, it´s awesome. I think for cheap and picturesque-while-centric, head to San Telmo. I grew up there, and it´s rather cheap, yet pretty, and touristy, and trendy, and close to Plaza de Mayo (the center of the city).
To go out and do stuff, you got loads. THere is plenty of free stuff going on, if you grab the college-student type magazines and newspapers, you will find craploads. If you want to pay, there is also stuff too. Remember that at most restaurants that have tango shows, having a foreign passport means paying 40 dollars instead of 40 pesos. Try to blend in! lol
Clubs, there´s loads if you want to go out, and for all tastes. You have the ever awesome Pacha, then Opera Bay. Amerika is good too, and for 20 pesos it´s ALL YOU CAN DRINK! it´s like the buffet of nightclubs. And a bunch of smaller places to go and listen to everything.
So good luck, and if you need anything else, let me know. I hope this helped!