The time differences in the short tri distances (like sprints) are much shorter than the longer races. Thus, if you really want to be competitive and shoot for a podium/1st place finish I'd recommend getting a tri top or (assuming the rules allow it) go shirtless.

A bicycle jersey would work in a pinch like sirious above mentions but unless you have one that is race cut and actually fits you like it was intended you'll probably be pretty annoyed with how baggy it feels during the swim.