If you've ever used exception handling in another language, it's not too unfamiliar in Perl. You can use eval to catch an exception thrown with die:

# code to open ERRLOG here
eval {
    die "There's no way around this!";
};

if ($@) {
    print ERRLOG "Caught: $@!\n";
}
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As for the directory deletion and warning, your code will always log a successful deletion, even if it fails. Try this:

if (rmdir ($_)) {
    open(LOG, ">>$logfile") or warn "discarding logfile output\n";
    print LOG "Directory: $_ - has been deleted.\n";
    close (LOG) or warn "Can't close: $!";
} else {
    warn "Cannot delete directory: $_ ($!)";
}
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