Hello,
I have been living in Bratislava for a few days now so I can't tell much about real cost of living (and I think it depends on personal preferences) but I am currently looking for a long-term rental so I can share my experience in this area.
Apartments in City Center are expensive and they are located in the very old buildings, even if they are completely refurbished so I wouldn't recommend this option. From almost any point within the city limits you can go to the city center by bus/tram/trolley in less than 25 minutes (at least according to schedules). Monthly public transport pass is 27EUR, cab ride is 5EUR.
You can find single apartment with utilities around 30 m^2 from 350 EUR but if you want something a little bigger and in good condition you should be prepared for 450-500 EUR.
3 beedroom apartments suitable for sharing can be found for 800-900 EUR but I didn't look for this kind of accomodation.
According to calculator you will get a 1 042 EUR net plus some meal vouchers which as far as I know are mandatory (3.15 EUR / workday). Average lunch in the city center is 5.5EUR. Prices of groceries you are going to use you can assess by looking at Lidl, Tesco or Billa websites (most common markets here.
I hope this helps.
Kind regards,
Chris