For HTTP requests, there's a Client-Peer header that you can access in the response...

use LWP::UserAgent;
use HTTP::Request::Common;

my $url      = 'http://www.perlmonks.org/';

my $browser  = LWP::UserAgent->new;
my $response = $browser->request( GET $url );

if ( $response->is_success ) {
  my($ip,$port) = split /:/, $response->header("Client-Peer");
  print "Connected to $ip on port $port\n";
} else {
  print "Couldn't get $url: ", $response->status_line, "\n";
}
[download]

... although it dissappeared briefly before (and then re-appeared with) LWP 5.66, so you may need to upgrade, downgrade or hack LWP::Protocol::http if you find it's absent from the version you have installed.

    --k.