You're looking at about 1600PLN/month + elec/gas/water for a 50sqm, newly renovated flat in a 1930s building, 10min walk to the Rynek
I have just found an apartment which is 3 tram stops from the centre (near Plac Inwalidow) which fits the above criteria specified by 'inkrakow' but the price is 1750 PLN per month + bills. I have been looking for a few weeks, and certainly it is hard to find anything decent for less than 2000 PLN per month in the current market