I'll first note that <pre> is your friend (but yay, view-source).

$time_string =~ m/^2013-03-[23][0-9]/
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would be one way to do it. Though I think I might prefer

my (undef,undef,undef,$mday,$mon,$year)=gmtime($mtime);
if ($year == 2013 && $mon+1 == 3 && 21 <= $mday)
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It's a hard call, what with having to deal with zero-based $mon

Original code below:

use strict;
use warnings;
use Net::FTP;
use Time::Local;
use File::Listing qw(parse_dir);
use POSIX qw(strftime);


my ($host, $user, $passwd) = ('xxx.xxx.xxx.xxx', 'abc', 'abc');
my $dir = '/abc/def/ghi';
my $ftp = Net::FTP->new($host)
    or die qq{Cannot connect to $host: $@};
$ftp->login($user, $passwd)
    or die qq{Cannot login: }, $ftp->message;
$ftp->cwd($dir)
    or die qq{Cannot cwd to $dir: }, $ftp->message;
my $ls = $ftp->dir();
$ftp->binary();
foreach my $entry (parse_dir($ls)) {
    my ($name, $type, $size, $mtime, $mode) = @$entry;
    next unless $type eq 'f';
    my $time_string = strftime "%Y-%m-%d", gmtime($mtime);
    if ($time_string eq '2013-03-29') {
        print "File $name has an mtime of $time_string\n";
        $ftp->get($name)
                or die "get failed ", $ftp->message;
    }
}
$ftp->quit;
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