I have seen several places where meters/branch circuits were not connected correctly.
One, an old apartment in San Francisco (decades ago)--the electric gate was connected to a co-worker's meter.
The second, an electrician wired a small commercial building for large A/C (when computer's needed to be kept cold to be reliable).
My suggestions--turn off/unplug everything in your apartment and see if the meter is still spinning. If it is, now start turning off each breaker (or unscrew each fuse) until you find the branch circuit that, when disconnected, stops the meter from running. Leave that branch circuit disconnected and restart everything else and see if all of your things are running correctly. Contact the landlord/owner about problem.
If your meter stops spinning when everything is turned off--you might have to work harder to find the problem.
If you are being billed for about 1,000 kWhr per month more than you should be using--that works about to about a 1,400 watt load running 24x7.
Your meter should be spinning pretty obviously with that amount of load (and everything else off)--if it is not spinning, I would talk with the power company and ask them if they can check the meter and see if it has any problems.
Also, check the meter readings with the bill... It is not unknown that a meter reader could have written the numbers incorrectly and the "computer" adjusted the reading to what it thought made sense...
Another way you can have problems... Example, you meter is read on the 15th, but you move in on the 1st and get new service. The old customer uses 5x the power that you do. The utility does not send anyone to read the meter on the 1st when you move in--but does their normal reading on the next 1st. Now, you have 2 weeks of the old tenant, and 2 weeks of yours. The utility sends 1/2 the bill to you and 1/2 the bill to the old tenant. You get screwed because the old tenant used 5x the amount of power vs your usage--but instead of the old tenant getting 5/6's of the kWh and you getting 1/6 of the billing--you each get 1/2.
Both of the above happened to me when I got my new service connected (old tenant used 5x+ my power), and when I went solar PV Grid Tie (computer "auto-adjusted" the meter reading)... My meter does run backwards (because of solar PV Grid Tie) and during much of the year I generate more power than I consume. In my case, the computer thought that the meter reader made a mistake and simply added 1,000 kWhrs to my bill to make up for the "error" (real problem was that they did not acknowledge the cross over between solar going live and a new TOU meter I had installed under the solar program).
Lastly, I have also seen issues where a meter is difficult to read (behind locked gate, covered, dirty, etc.). And when the service is updated for the new customer, the estimated usage is "adjusted" to match with the actual meter reading--sometimes giving a big surprise when the meter is actually read.
It will probably take some detective work on your part, and you may need to show them a copy of the old bills (or give them your old address/account number if same utility) to get them to adjust your first bill.
Hope this gives you some ideas to check out.
-Bill