For starters, very few servers will allow you to delete; hence, "delete" $ftps->delete($file). Also, if the timestamp is being mangled, that's because, by default, Net::FTPSSL doesn't preserve timestamps. You'll need to add PreserveTimestamp to the new method. Here's what I tried:

#!/usr/bin/perl

use strict;
use warnings;
use Net::FTPSSL;

use constant HOST => 'ftp.perl.org';
use constant DIR  => '/pub/CPAN';
use constant REMOTE_FILE => 'README';
use constant LOCAL_FILE  => 'README';

my $ftps = Net::FTPSSL->new(HOST,
           PreserveTimestamp => 1,
           Port  => 21,
           Debug => 1,
           Croak => 1,
           Trace => 1);

$ftps->login('user', 'pass');  
$ftps->cwd(DIR);     
if ( $ftps->get(REMOTE_FILE, LOCAL_FILE) ) {
   open(MYLOG, '>', 'Mylog.txt');
   print MYLOG time() . " LOCAL_FILE was downloaded from HOST\n";
} 
else { 
   print "There was a problem.\n";
}
unlink(LOCAL_FILE);
$ftps->quit();
[download]

Update: Here's an example using Tie::Scalar::Timestamp

#!/usr/bin/perl

use strict;
use warnings;
use Net::FTPSSL;

use constant HOST => 'ftp.perl.org';
use constant DIR  => '/pub/CPAN';
use constant REMOTE_FILE => 'README';
use constant LOCAL_FILE  => 'README';

my $ftps = Net::FTPSSL->new(HOST,
           PreserveTimestamp => 1,
           Port  => 21,
           Debug => 1,
           Croak => 1,
           Trace => 1);

$ftps->login('anonymous');  
$ftps->cwd(DIR);     
if ( $ftps->get(REMOTE_FILE, LOCAL_FILE) ) {
   use Tie::Scalar::Timestamp;
   $Tie::Scalar::Timestamp::DEFAULT_STRFTIME = '%H:%M:%S';
   open(MYLOG, '>', 'Mylog.txt');
   tie my $timestamp, 'Tie::Scalar::Timestamp';
   print MYLOG $timestamp . " LOCAL_FILE was downloaded from HOST\n";
} 
else { 
   print "There was a problem.\n";
}
unlink(LOCAL_FILE);
[download]