Submit to arXiv with a different order of authors
I don't think there are any rules against having a different author ordering on the arXiv version, but I'd recommend against trying this.  The author ordering on the published paper would be regarded as "official", and doing things differently on the arXiv would certainly not be considered to demonstrate the real first authorship.  Instead, it could easily be misinterpreted:


People may assume you were careless about author ordering in submitting to the arXiv, which wouldn't be good for your reputation.
Worse yet, they may assume you were dishonestly changing the author ordering, which would be a disaster for your reputation.
The best case scenario is that they would assume the author order changed between the preprint and the published paper, because the authors changed their minds about who should be listed first.


The only way to avoid these dangers would be to add an explicit note explaining the change, but I assume that's out of the question.  After all, if you deserve to be first author on the arXiv paper, then you should have been first author on the published paper as well, so such a note would just raise ethical questions about the published authorship.

It sounds like you should have been first author all along, since author ordering should not be determined based on how much credit the university gives.  If it is still possible to fix the author ordering before publication, that could be the best approach, but the arXiv is not a good way to fix it after the fact.