if I sue the author of the libelous [sic] content and claim this as part of my damages, will I have to directly show that my rejection was due to the school believing the content?
Yes.
Furthermore, how could this be proven?
Ideally, by having evidence that this was the reason from the school that rejected you. This could be testimony from someone who knows or written evidence of how your application was assessed. Both of which could be subpoenaed and both of which might not exist. Further, if the basis is unlawful under anti-discrimination law, the school has an incentive to pretend that they used a different basis.
Alternatively, if you could demonstrate a clear change in the tone of correspondence from the school before and after the allegedly libellous post(s) were made or came to the attention of the school.
However, ...
You don't seem to have suffered any damage from this rejection as your application fee was refunded. Showing that this particular rejection will affect your future earnings would be ... problematic, especially in light of the fact that it appears that graduate education was not so important to you since you only made one application.