I have seen some apartments on a web page, just google. They are not much to look at, at least not from what I have seen, or not what you would get in Kiev or Odessa. but they are right near the Moscow store on Lenin street. There is a club right by there and an open market and a cheap liquor store in a back alley. You will need to bring a local for the last two.
By the way that is a lousy time to be in Mariupol. Not cold enough or snowy enough for winter fun, Just enough for cold feet especially when they turn the heat off at night.