Take a peek at File::Listing's parse_dir function. You should be able to grab the file size by using that. Untested:

use File::Listing qw(parse_dir);

# your ftp code here.
my @dirlist = $ftp->ls("-l");
for (parse_dir(@dirlist)) {
   ($name, $type, $size, $mtime, $mode) = @$_;
   next if $type ne 'f'; # only want plain files
   print "Size of $name is $size\n";
}
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#!/usr/bin/perl
use strict;
use Net::FTP;

my $user = "someuser";
my $passwd ="password";;
my $host = "someFTPserver.com";
my $file = "somefile.txt";
my $ftp = Net::FTP->new($host) or die "$@";
$ftp->login($user, $passwd) or die $ftp->message;
my $dir = "/home/somedir/mydir/dest";
$ftp->cwd($dir) or die "Can't cwd to $dir\n";
my $remote_size =  $ftp-> size($file);
print ("The source file size is :", $remote_size , "\n");

$ftp->quit;
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