The cgiapp_prerun() method is the natural place to do authentication (teardown is for cleanup e.g. closing database connections).
The following is from CGI::Application's documentation (see prerun_mode()):

  # In WebApp.pm:
  package WebApp;
  use base 'CGI::Application';
  sub cgiapp_prerun {
        my $self = shift;

        # Get the web user name, if any
        my $q = $self->query();
        my $user = $q->remote_user();

        # Redirect to login, if necessary
        unless ($user) {
                $self->prerun_mode('login');
        }
  }
[download]

Once you've done something like the above in your MyAuth module, your other modules can subclass MyAuth to reuse the authentication code.
Some useful nodes: Re: Re: Re: Why CGI::Application?, Re: Breaking up Large Modules..