My biggest experience outside of Buenos Aires is Cordoba, but I've only spent a few weeks there, mostly on business. It's a nice place, clean, the attitude of the people different from here substantially (and their accent a bit harder to understand!). It is considerably cheaper than here.
I don't know about finding places to stay. I would assume that you could probably find apartments on short term, but I doubt they will be as plentiful as here, considering that BA is a center of tourism on a big scale and renting temporary apartments is a pretty big business here. It may be more difficult to find apartment shares there.
I would think work would be more difficult as well, but I have no empirical evidence to back up my opinion.
I have some Argentine friends who live here (about 6 years now), but are originally from Cordoba. They moved back to Cordoba from here about two years ago because they said they didn't like BA. They lived in Cordoba for a year, then returned to BA. They told me they realized BA was better than Cordoba in certain ways that they had become accustomed to. For example, the transportation in Argentina of goods is not very good - many of the railroads are unusable for shipping stuff and things are carried by trucks instead. This causes shortages of certain "luxury" goods like furniture, the biggest example they used. They couldn't find any furniture that was to their taste, the only being what was produced locally at any kind of reasonable price or availability. Special things had to be ordered and shipped specially taking a long time. Services weren't as good and they had a much harder time finding a job in the IT industry.
Much of that particular point may not mean too much to you though.
There are ways to live cheaply here, though not necessarily comfortably (and at times not particularly safe!). My family (outside of the three who live with me) live in "Hoteles" for example. The prices on those can range from 500 pesos to 1500 pesos a month. At the lower end are bedrooms with no windows and a shared bathroom and kitchen down the hall (no hot plates allowed in the rooms) to a private bathroom with a small area where you can use a hotplate.
I've seen hoteles (they are like "pensiones") in good parts of the city as well as bad.
I've seen a lot of people on this forum looking for and advertising apartment shares which are almost certainly better and can end up costing about the same as some of the higher end hoteles, if you can find one. Or you may find a family or a couple renting out a room.
Food-wise - most Argentinos I have known or had experience of eat very meagerly and cheaply (and not necessarily healthily) to save money. Most of my younger sister-in-law's schoolmates have typical meals of pasta with a little oil and something like oregano to season it, for example. Or they buy pizzas from cheap places when they get together. She doesn't like pasta or pizza, so she always comes back hungry from those outings
On the subject of work, My wife's friends from Paraguay who come here usually work one of two ways: live-in or "de retiro" maid (de retiro meaning they work set hours and live elsewhere), or in lavanderias. I only mention these jobs because it's what I've seen on my end as far as easy jobs to get for people with limited contacts working in the black, but they are not well-paying. The friends who work as live-in make something like 1500 pesos a month, those working de retiro make something like 1800-2000 a month. Either one of the maids work long hard hours usually, although at least the de retiros have an end to their day. The ones working in the laundries do better I think, as they typically work 5 and a half days a week, about 9 hours a day, and make around 2500 pesos a month.
Obviously it would be better to work teaching English or something if you can find it - it would probably pay more, although I don't know how steady it is. Also, with the economy the way it is, and with as many people as I've seen with the same sort of idea, how easy it is to find jobs like that. Certainly possible, but unknown how long to actually find something to offset your costs quickly.
