LMAO Eileen, you party animal you!

Im sure Kate means prams on the sydney trains, so i would say no dont bother folding it up, once you get the pram on the train, someone might be kind enough to help you, just stay in the main section there which has heaps of room, there are seats along the windows, or just stand if its busy. Peak hour would be crazy, but if you're travelling during the day, say after 10 ish you shouldnt have a huge problem. Not worth folding the pram to squeeze down the stairs and into a booth type seat.
The bus station isnt far from central but it is a bit of a walk. I dont know if you can catch a train to the Uni? I think its only buses..