I've lived in Hyde Park for the past 5 years. The cheapest 1 bedrooms are going to be $800+. My boyfriend and I currently pay $885 for a 540 square foot 1bed/1bath. You can find places a bit cheaper than this if you go south of 60th street, but it's not the safest area.
I don't know much about Bridgeport, other than the fact that a coworker lives there and has an easy commute to Hyde Park on the #4 bus.
If you have any questions about Hyde Park or the rental market here, let me know.