my $last = '';
open FILE, "<whatever.txt" or die "Oops $!\n";
open TMP, ">tmp.txt" or die "Oops $!\n";
while (<FILE>) {
    $last =~ s/this/that/ if $_ eq $some_next_line_bit;
    print TMP $last;
    $last = $_;
}
print TMP $last; # print the very last line
close FILE;
close TMP;
rename "tmp.txt", "whatever.txt";
unlink "tmp.txt";