Let me say again that this is a really good question--Why would the fellow who erected an official residence for the tribunes not be a former tribune himself? Strangely, no one has addressed this question before (at least not that I can find).
Best I can tell, after his quaestorship, Cato had one of two offices for which he could run: the tribuneship and the plebeian aedileship. I know of no evidence suggesting that either route was a shorter path to a praetorship, but at least in Cato's case, his aedileship was extraordinarily successful, so much so that he was permitted to run for a praetorship immediately (i.e., without the customary delay between offices). This dispensation appears to rest on the two main achievements of his aedileship: the restoration of the Plebeian Games (again, supporting the idea that he was proud of his plebeian status) and his vigorous re-organization of what passed for a police force in Rome. One is tempted to infer that Cato chose the plebeian aedileship in order to restore the Plebeian Games, but of course that's impossible to know.
I'll keep an eye out for a better answer, but that's the best I've got for now. (BTW, in researching the answer to this question, I was surprised by how sympathetic Mommsen was to this Cato.)
Thank you, MPC. This is something more to think about.