Apparently I can do it with Apache 1 as well.. Running httpd -v gives me:

Server version: Apache/1.3.27 (Unix)  (Red-Hat/Linux)
Server built:   Mar 15 2005 14:49:39
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The final handler I used (not the one given for the thread question) was the following for handling CGI scripts:

<Perl>
{
    package SessionLogAndHandle;
    use CGI::Session;
    use Apache::PerlRun;

    sub handler {
        my ( $r ) = @_;

        {
            my $s = CGI::Session->new();
            my $authuser = $s->param( 'AUTHUSER' );
            $r->user( $authuser ) if ( $authuser );
        } # I want $s to go out of scope

        return( Apache::PerlRun->handler( $r ) );
    }
}
</Perl>

<Directory "/var/www/myproject/bin">
        Options +ExecCGI
        SetHandler perl-script
        PerlHandler SessionLogAndHandle
        Allow from all
</Directory>
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Update: removed superfluous session flush because I wasn't modifying the session.