I use mod_fcgid with a Plack app and it just works (and it works better than mod_fastcgi may I add, for whatever reason). You seem to be having an Apache configuration issue rather than a Plack issue here though?
I'll admit it took me a little while to get my setup working, mostly having to do with understanding what path to put in a Rewrite rule location in a .htaccess file and nothing to do with Plack itself.
What doc are you using? I started with the Dancer deployment perldoc and adapted from there:
. A quick Google search showed that similar documents exist for Catalyst and Mojolicious.