The lowest priced listing in Lincoln Park right now is $1250, for a studio. You're better off farther north. There's a place on Southport for $975. That's the lowest priced listing in Lakeview.There may places in unadvertised buildings. Have you been scouring Craig's List?
There isn't anything in the MLS in Lincoln Park or Lakeview in that price range. Rental prices are at an all-time high. You'll probably need to go farther north, like Edgewater or Rogers Park, to stay on the Red Line in that price range.