#!/usr/bin/env perl

use strict;
use warnings;

my $wanted_sub = 'Subroutine2';
my $pod_string;

{
    local $/ = "\n=head1 ";

    while (<DATA>) {
        if (/\A $wanted_sub $/mx) {
            ($pod_string = $_) =~ s/\A .+? ^ ( .+? ) =cut .* \z/$1/msx;
            last;
        }
    }
}

print $pod_string;

__DATA__
package MyModule;
=head1 Subroutine1
My POD for Subroutine1
=cut
sub Subroutine1
{
...
...
}

=head1 Subroutine2
My POD for Subroutine2

=head2 Dummy Heading 2

=over 4

=item whatever

qwerty

=item blah

asdfgh

=back

Some text

=cut
sub Subroutine2
{
...
...
}
1;