Because the function is only returning one item from the list there is no reason to iterate through and sort the entire list:

sub _file_containing {
    my ( $self, $id ) = @_;

    opendir my $dir, $self->{ update_dir } or return;

    my $ret;
    while ( my $file = readdir $dir ) {
        next if $file !~ /^\d+$/ || $file > $id;
        $ret = $file if $ret < $file;
        }

    return "$self->{update_dir}/$ret";
    }
[download]