I agree with Rivendell. I would never use "postular" except if I'm talking about someone who is defending a theory: "Miguel Servet postulaba que exitía una doble circulación sanguínea".
I would say "He pedido/solicitado la admisión en la universidad X", because when you apply you still don´t know if your application is going to be accepted. After the acceptance, you pay and send all the necessary papers: this is "matricularse" (me he matriculado...).
And if I am applying for a job I would also say "he echado un CV a un puesto de trabajo". It sounds more informal, but it is the most usual option.