Even the inner city is full of good supermarkets (high quality, cheap, huge selection), and here is how to find them

Adresses and maps of all supermarket stores in each inner-city district:

Södermalm
New town
Gamla stan
Kungsholmen
Vasastan
City
Östermalm

Extra directions to the cheaper supermarket stores:

Willy:s:
Götgatan 79 in district Södermalm. Go to the subway station Skanstull and choose the exit that says Blekingegatan. Once on the street (which is Götgatan) you just have to cross it, and number 79 will be right there.) Open every day until 22.00.

Lidl:
Magnus Ladulåsgatan 3 in district Södermalm. Take bus number 4 (towards Gullmarsplan/Radiohuset) and get off at the stop Södra station, or take the commuter train towards Södertälje or Västerhaninge/Nynäshamn and get off at the station Stockholm Södra and then choose the exit that says Rosenlunds sjukhus. Magnus Ladulåsgatan is a side street to the street where you are standing (Rosenlundsgatan). Opening hours: Mon-fri 09-20, sat 9-19, sun 10-19.

Sveavägen 57-59 in district Vasastan. Go to the subway station Rådmansgatan. The exits leads to the street Sveavägen. Opening hours: Mon-fri 09-22, sat and sun 09-21.

S:t Göransgatan 70-72 in district Kungsholmen. Go to the subway station Fridhemsplan and choose the exit that leads you to S:t Eriksgatan, then walk straight a little bit. S:t Göransgatan is a side street to the street where you are standing (S:t Eriksgatan). Opening hours: Mon-fri 09-21, sat and sun 10-20.

Folkungagatan 51 in district Södermalm. Go to the subway station Medborgarplatsen and choose the exit Folkungagatan. Once on the street (which is Folkungagatan) you turn left and walk to number 51.