I always carry the 3 B's : Bath towel, Baby wipes, and bagels. If you don't have shower facilities then just using a restroom can help you clean up quite a bit. Also I always wear a cotton undershrit and a biking shirt that wick the moisture away. If you take it at a leisurely pace it should not be too bad and as trebuchet03 states you can always take spare cloths.
I agree also that you can probably find a good MTB or road bike for under 100 bucks or so that will get the job done,especially in a college area . It won't take so much effort on 26" or 700cc wheel as the BMXer.
Or you can always load up your bike and drive part of the way. Ride the rest until you find the sweet spot(better then driving the whole thing) where you not overdoing. Most of the time, unless it windy, you always have a headwind to help cool you off.
Or you can just buy a gallon of Old Karate and use liberally
