The short answer is that you need to use $nextLine = <INPUT> to get the next line after you match, then use last to leave the loop.

Here's something I patched together:

#!/usr/bin/perl -w

# Next After Match .. PM node 418383

my $filename = shift;

open ( INPUT, $filename ) or
  die "Unable to open $filename: $!";

my $string = shift;

my $nextLine;
while(<INPUT>) {
  if ( /$string/ ) {
    $nextLine = <INPUT>;
    last;
  }
}

close ( INPUT );

if ( defined ( $nextLine ) ) {
  print "INFO: $string found in file $filename, next line is:\n";
  print $nextLine;
} else {
  print "INFO: $string not found in file $filename.\n";
}
[download]

When I run it on itself I get the following:

[foo@bar dev]$ perl -w nam.pl nam.pl print
INFO: print found in file nam.pl, next line is:
  print $nextLine;
[foo@bar dev]$ perl -w nam.pl nam.pl whoosat
INFO: whoosat not found in file nam.pl.
[download]

Seems to work.