Sure. You just want to add an argument to your get statement with the new directory:

#!/usr/bin/perl -w

use Net::FTP;
use CGI qw(-oldstyle_urls :standard);
use CGI::Carp qw(fatalsToBrowser);
use strict;

my $q = new CGI;


my $host="myhost";
my $dir = "/home/docs/test/";
my $new_dir = "/home/docs/new_test";
my $login = "xyxyxy";
my $pw = "uyuyuy";

my $ftp = Net::FTP->new($host) or die "Cannot connect to $host: $@";
$ftp->login($login,$pw) or die "Cannot login to $host as $login: $@";
$ftp->cwd($dir) or die "Cannot change working directory: $@";

my @files=$ftp->ls or die "Cannot list current directory: $@";

foreach my $x_files(@files) {
    $ftp->get($x_files, "$new_dir/$x_files")  
        or die "Can't fetch $x_files: $!\n";
}


$ftp->quit;
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