If you want to record the contigs, don't save just the palindromes in the hash, but group them by contig first:

#!/usr/bin/perl
use warnings;
use strict;

open my $IN, '<', 'sample.txt' or die $!;
my $pat = qr/^(Contig *([0-9]*))\s/;
my $count = 0;
my @regexes;
for my $n (5 .. 20) {
    my $re = qr /[CAGU]{$n}/;
    $regexes[$n-5] = $re;
}

my %palhash;
my $contig;
LINE: while ($count < 1000) {
    my $line = <$IN> ;
    defined $line or last;
    $contig = $line if $line =~ /$pat/;
    ++$count;
    for my $value (@regexes) {
        my $start = 0;

        while ($line =~ /$value/g) {
            my $endline = $';
            my $match = $&;
            my $revmatch = reverse($match);
            $revmatch =~ tr/CAGU/GUCA/;
            if ($endline =~ /^([CAGU]{0,15})($revmatch)/) {
                $start = 1;
                my $palindrome = $match . "*" . $1 . "*" . $2;
                $palhash{$contig}{$palindrome}++;
            }
        }
        next LINE if $start == 0;
    }
}
close $IN;
for my $contig (keys %palhash) {
    print $contig;
    while (my ($key, $value) = each (%{ $palhash{$contig} })) {
        print "$key => $value\n";
    }
}
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I changed some parts of the code as well, e.g. 3 argument open or die, lexical filehandles, don't process if the input is short, no goto. I don't understand the requirements in detail, but moving away from $' and $& would be a good idea, too.