Something like this would probably do:

use strict;
use warnings;

my $temp = "DATA\nFile\nHello\nAnother line\n";

my $found;

if( $temp =~ m/File\n(^.*$)/m ) {
    $found = $1;
    print $found, "\n";
}
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...I like the anchors in this regex, but there are a lot of other ways to craft the RE. Or you could to it this way instead:

my @lines = split /\n/, $temp;
foreach my $idx ( 0 .. $#lines ) {
    next unless $lines[ $idx ] =~ m/File/;
    $found = $lines[ $idx + 1 ];
    print $found;
    last;
}
[download]