$ip is your object, not $ARGV. Change
$ARGV->reverse_ip()
to
$ip->reverse_ip()

Other problems:

• Don't use $a, it's a special variable.
• Don't use $b, it's a special variable.
• Don't use $ARGV, it's an even more special variable.
• You fetch from $ARGV[0] when you mean to fetch from $ARGV or $a.
• You're using the dangerous indirect method call notation (new CLASS ARGS) instead of the direct methods (CLASS->new(ARGS))
• You needlessly duplicated your use statements.
• You don't handle the case where no names are associated with the IP address.
• You don't handle errors.

Fixed:

use strict;
use warnings;

use Tk;
use Net::IP;
use Net::DNS;

my $ip_field;
my $name_field;

my $resolver = Net::DNS::Resolver->new()

sub rlookup {
    my $ip = Net::IP->new($ip_field, 4);
    my $answer = $resolver->query($ip->reverse_ip(), 'PTR');
    my @names = $answer->{'answer'};
    if (@names) {
       $name_field = $names->[0]->{'ptrdname'};
    } else {
       $name_field = "[no associated name]";
    }
}

{
   my $mw = MainWindow->new;
   $mw->title("IP Lookup");
   $mw->Label(-text => "Enter an IP Address")->pack;
   $mw->Entry(-textvariable => \$a)->pack;
   $mw->Button(-text => "Lookup", -command => \&rlookup)->pack;
   $mw->Entry(-textvariable => \$b)->pack;
   MainLoop;
}
[download]

It still doesn't handle errors.

The original code from http://www.stevekallestad.com/blog/reverse_ip_lookup_using_perl.html is:

#!/usr/bin/perl
use strict;
se warnings;
use Net::IP;
use Net::DNS;

my $ip = new Net::IP($ARGV[0],4);
print $ip->reverse_ip()."\n";
print "Resolving ...\n";
my $res = Net::DNS::Resolver->new;
my $answer = $res->query($ip->reverse_ip(),'PTR');
my $namer = $answer->{'answer'}[0];
print "PTR Name: $namer->{'ptrdname'}\n";
[download]

@ARGV is a special variable that contains the parameters passed to the script. In the script you copied the ip was passed as a command line option. In your case it is in the variable $a, so you must use this variable in the call to new Net::IP like this:

my $ip = new Net::IP($a,4);
[download]

You better don't change the @ARGV variable at all.

BTW, when you write:

my $ARGV = $a;
[download]

You are assigning the scalar variable $ARGV. And when you write:

my $ip = new Net::IP($ARGV[0],4);
[download]

You are using the array variable @ARGV (its first element)

citromatik