Whether there is something wrong with your A/C unit or not (check the fridge, too! Defrost!), I can give you a few general tips to conserve in a 1br apartment:
1) Get a programmable thermostadt and set it for consistent temps, day and night.
2) Set it as high as you can stand it and add a few small fans; as long as the air is dry, fans are remarkably effective in making you feel cooler.
3) Set it a few degrees higher during the day when you're not home. But DON'T fall for the temptation to set it way higher, or else you'll waste a lot of power cooling it back down to max comfort level when you get home. I set mine to 74 for sleeping (11:30pm-7am), 78 when I'm home (7-8am and 5-11:30pm), and 82 when I'm at work (8am-5pm).
4) Keep blinds/drapes completely closed during the day. Yes, turning on lights in the daytime might seem more wasteful, but the sunlight coming through the windows and hitting the floor, walls, and furniture radiates heat throughout the room. Just keeping that light out can lower the temp in the apartment by 5 degrees!
5) Turn off your computer, monitor, printer, scanner, and speakers off whenever you're not using them.
6) Replace all of your light bulbs with low-consumption flourescents, especially floods. Get rid of any hallogen torch lamps - they not only use a lot of power, but generate heat, which makes your A/C work harder.
7) Don't cook with the oven or stove in hot weather. Use the microwave for as much as you can. Yes, it uses electricity, but it also puts a lot less heat into the apartment that the costly A/C must then cool off. I also use my Foreman grill a lot; I get some great meals without heating up the kitchen much.
8) Open the bedroom and bathroom doors and use fans to circulate the cool air throughout the apartment to eliminate hot spots, instead of just cranking the thermostadt to feel cooler. Consistency and even cooling will help you bear the heat better and reduce your need for cooling.
9) Any time the temp and humidity outside drop to comfortable levels, turn the A/C completely off, open the windows, and run several fans to circulate the air through the apartment. Close the bathroom door so your hallway will promote smooth airflow, use the bedroom as the intake and the living room as the exhaust, and make all your fans push the air in one direction. Circulating the outside air through the apartment for just a few hours a night without the A/C can add up to significant savings over a week.[NOTE: In my area, this is possible only during the spring and fall months because of humidity, but it saves me tons of money during those months.]
10) Turn stuff OFF when not using it - lights, stereo, computer equipment, video games, even the cable box. Just putting stuff in stand-by mode can reduce its consumption by 90%, which can add up if you do it to a lot of devices throughout the apartment. Put the computer in stand-by while cooking and eating dinner or watching TV.
All of this stuff is just gap-filler, though. The largest consumer in your apartment is the A/C, followed by the fridge. The most effective ways to lower your bill will be those that address the A/C and fridge - either directly (i.e. use it less) or indirectly (don't make more heat for the A/C to get rid of).
Good luck.