Jumbo is typically one of the most expensive places to buy food, but it often has good quality. Even though Disco is a part of Jumbo, it's cheaper, even if sometimes more expensive than others. I have Disco a block away from my apartment at Esmeralda and Libertador, a somewhat snooty area, and it's cheaper than Jumbo in Palermo. They are smaller and don't have quite the same availability, but you could probably shave a couple hundred pesos at least off your monthly budget by doing some of your shopping there.
All the chains, as far as I know, deliver. Find a cheaper place except for the things you can only get at Jumbo.
For meats and veggies, you should find a good shop that specializes. Find a good place near you if you can. It should be cheaper than what you see in the super market, or maybe you're in the wrong neighborhood to get it nearby.
I can find good whole lomos (1.6 kilos on average), for example, at 48 pesos a kilo (yes, seriously), where it's somewhere around 120 pesos a kilo where I live, in the supermarket, for a trimmed lomo (less than a kilo often).
There is a vegetable stand near where I go to get meat sometimes that has good priced fruits and vegetables, about 1/2 the price of the supermarkets near me.
There's also a guy that comes around selling fruit. It's the first building I've ever lived in that had someone doing that - you might ask your portero if there's someone like that that comes around you're place.
You can knock a third to a half off your meat and veggie/fruit budget if you work at it. It's not always easy.
I have to go a ways sometimes, but I have a car. I go to Matadero and the edges of Constitucion usually for both meat and veggies. Sometimes, there's a Frigorifico that opens near us and sells good meats at a good price, but you never know when it will be open. Sometimes we go out to Wal-Mart north of the city (although there is a new one in Dot Baires Mall, but I don't much like it there) to stock up on drinks and cleaning supplies and such, even some half-assed glazed donuts.
This is part of living in Buenos Aires, finding your way around and finding the deals that are near you, or relatively convenient depending on your transportation limitations. It's one thing I like about having a car - the freedom to be able to take a half hour drive (if I'm lucky there's no strike blocking the road or I am within the small mid-day window when traffic on some roads isn't TOO bad!) to a place that's farther than that by far in time and effort by bus or subway, to find deals on daily consumables. Also, it's very nice to be able to buy 20 kilos of meat in one shot, enough for a goodly portion of the month, and be able to carry it home easily. Or go out to Puerrydon and the 25 de Mayo highway, where there are a ton of kitchen and restaurant supply places and buy things for the kitchen and easily take it back.
Without a car, you are constrained to what's close by, or how much time and effort you are willing to spend to make trips to places farther away.
