You can turn the $client_addr received from accept into a human-readable IP address using unpack_sockaddr_in and inet_ntoa (and then echo that back to the client, as sn1987a described in Re: Client Server). That said, Socket is a fairly low-level module; are you sure you don't want something else instead, say IO::Socket?

For instance, here's the server:

#!/usr/bin/perl

use warnings;
use strict;
use IO::Socket;
use Socket;

# use port 7890 as default
my $port = shift || 7890;

my $socket = IO::Socket::INET->new(
   LocalPort    => $port,
   Type         => SOCK_STREAM,
   Reuse        => 1,
   Listen       => 10,
)
   or die "Couldn't create server: $@\n";

while (my $client = $socket->accept()) {
   my ($port, $iaddr) = unpack_sockaddr_in($client->peername());
   $client->send(inet_ntoa($iaddr));
   $client->close();
}
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And here's the client:

#!/usr/bin/perl

use feature qw/say/;
use warnings;
use strict;
use IO::Socket;

# initialize host and port
my $port = shift || 7890;
my $server = "localhost";  # Host IP running the server

my $socket = IO::Socket::INET->new(
    PeerAddr    => $server,
    PeerPort    => $port,
    Proto       => "tcp",
    Type        => SOCK_STREAM
)
    or die "Can't connect to server: $@\n";

while(<$socket>) {
    say;
}

close $socket or die "Can't close socket: $!";
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Credit where credit is due: all this is pretty much directly copied from The Perl Cookbook, specifically Recipes 17.1 ("Writing a TCP Client"), 17.2 ("Writing a TCP Server") and 17.7 ("Identifying the Other End of a Socket". I really recommend that book, it's very handy.