Many years ago, Buenos Aires lost interest in big central food markets like the ones in Florence, or Seattle, or most cities in Europe.
Most have been gone for 20 years or more.
The main wholesale food market was moved far out into the suburbs, and is not friendly to individual buyers who want less than a case of anything.
The one mentioned above, Mercado Progresso,
http://www.mercadodelprogreso.com.ar/
on Rivadavia, mere steps from the Primera Junta Subte station, is one that remains. It is great, and the whole neighborhood around it has many more food stores and other interesting down to earth shopping.
The other one that is close in to the center of town is in San Telmo-
http://www.welcomesantelmo.com/san-telmo-guide/rmb3jqv6p9/
this one has antiques, second hand oddities, and food as well.
But each is a bit of a trip from Recoleta.
Parts of Recoleta are so upscale that the maids are sent farther afield to buy food- when you can rent to Prada and Louis Vuitton, the greengrocer probably wont pay the amount of rent you want. if you get a bit towards Barrio Norte, there are more corner stores, meat markets, and vegetable stands.
You will find you will have to try various stores, and come up with a mix of stores that fits your particular taste. But you should still be able to find most things you need within a few blocks.
Dan has a good list of food stores-
http://www.saltshaker.net/buenos-aires-food-drink
Where, more precisely, is the apartment you are renting?
we can give some recomendations if we know more. Up near the big library on Las Heras are more neighborhoody places.