If you don't have root access then you cannot really call it "my
server" can you?
Catalyst has a lot of dependencies and is not easy to install, not having root access will make it even more difficult, but if you can run cpan it should not be impossible to do it. You can perhaps try to install it under your home-share
However, that is only part of the job. You must then configure the webserver to run Catalyst and for that you will probably need higher authority than you have. Use your social skills on the webmaster to link Catalyst into the webserver.
A program should be light and agile, its subroutines connected like a string of pearls. The spirit and intent of the program should be retained throughout. There should be neither too little or too much, neither needless loops nor useless variables, neither lack of structure nor overwhelming rigidity." - The Tao of Programming, 4.1 - Geoffrey James