ear,
Beyond the sagely advice already provided, I wanted to mention a few things for further consideration.

First, your hash keys make it difficult to sort on them chronologically which presumably you would want to do:

for my $week ( map { $_->[0] }
               sort { $a->[1] <=> $b->[1] || $a->[2] <=> $b->[2] }
               map { [$_, split /\./] } keys %IssuesByWeek
             ) {
    # ...
}
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If you can, it would be better to use a ISO date string that can be sorted and will only require conversion one time and not every time you need to print results:

for my $key ( keys %IssuesByWeek ) {
    # New key in format of YYYY-MM-DD
    $IssuesByWeek{ get_sunday( $key ) } = delete $IssuesByWeek{ $key }
+;
}
[download]

If you can't do that, then perhaps you could change your data structure to something like $data{year}{week}{issue} = $count; then it will be much easier sorting.

The second thing is that it is generally a waste to sort a list to find the top or bottom of the list. There are routines known as water-mark algorithms that do this much more efficiently. Instead of sorting the entire list and then throwing away most of the work, I would suggest using something similar to:

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

my @list = map { int(rand 1000) + 1 } 1 .. 50;

print "Top 5:\n";
print "\t$_\n" for top_x(\@list, 5);

sub top_x {
    my ($list, $x) = @_;
    $x--;
    my @top;
    $#top = $x;
    for my $item ( @$list ) {
        next if defined $top[ -1 ] && $item <= $top[ -1 ];
        for my $id ( 0 .. $#top ) {
            $top[ $id ] = $item and last if ! defined $top[ $id ];
            if ( $item > $top[ $id ] ) {
                @top[ $id .. $#top ] = ($item, @top[ $id .. $#top - 1]
+);
                #splice(@top, $id, 0, $item), pop @top;
                last;
            }
        }
    }
    return @top;
}
[download]

Note that the commented splice will effectively do the same thing as the array slice. Depending on the size of the array, it may or may not be more efficient.