I modified your script, so that it outputs a little more information:

use strict;
use warnings;
use Time::Piece;

my $ctime = Time::Piece->new;
my $oldtime = Time::Piece->strptime('Dec 19 17:44:00 2016', '%b %d %T 
+%Y');

$oldtime = $oldtime - $ctime->tzoffset;

sub showtime { shift->strftime }

print "The current time is ", showtime($ctime), "\n";
print "The old time is ", showtime($oldtime + $ctime->tzoffset), "\n";

my $halfago = $ctime - 1800;

print "Half an hour ago is ", showtime($halfago), "\n";

if ($oldtime > $halfago)
{
    print "Old time is within a half hour\n";
}
else
{
    print "Old time is outside a half hour\n";
}
[download]

The parameterless form of Time::Piece's strftime includes time zone name. My output was:

The current time is Di, 20 Dez 2016 10:09:19 Mitteleuropäische Zeit
The old time is Mo, 19 Dez 2016 17:44:00 UTC
Half an hour ago is Di, 20 Dez 2016 09:39:19 Mitteleuropäische Zeit
Old time is outside a half hour
[download]

Of course, since I didn't update $halfgo, it is really outside, but it illustrates davido's point.