That's correct: you cannot add something to the beginning of a file without rewriting the whole file.

As it seems, the code does not implement one of the required tasks: 'then remove those 3 lines from daily.txt'.

Of course, this can be fixed:

#!/usr/bin/perl -w

my $daily="daily.txt";
my $weekly="weekly.txt";
my $lines=3;
# alternatively, read the names from the command line:
# my($daily,$weekly,$lines)=@ARGV;

# read the daily file:
open DAILY, $daily or die "open $daily: $!";
my @daily=<DAILY>;
close DAILY;

# read the weekly file:
open WEEKLY, $weekly or die "open $weekly: $!";
my @weekly=<WEEKLY>;
close WEEKLY;

# if the last line of @daily has no \n, this line will
# be merged with the first line of @weekly.  We are
# paranoid and make sure it is there!
$daily[-1]=~/\n$/ or $daily[-1].="\n";
# Well. That's good for Unix, not for Windows. You would
# have to change "\n" to whatever is appropriate.

# move the last $lines lines from the end of @daily to
# the beginning of @weekly. Note:
# splice(@daily, -$lines) removes and returns the last 
# $lines lines from @daily;
# unshift @weekly, @some_lines adds some lines at the
# beginning of @weekly.  Together:
unshift @weekly, splice(@daily, -$lines);

# Write back the weekly file:
open WEEKLY, ">$weekly" or die "open >$weekly: $!";
print WEEKLY @weekly;
close WEEKLY;

# Write back the daily file:
open DAILY, ">$daily" or die "open >$daily: $!";
print DAILY @daily;
close DAILY;
[download]

You might be interested in the splice and unshift manpages for documentation of the main step of this program.

By the way: It is possible to append to the end of a file by opening it with open FILE, ">>$name".

With the truncate function, you can truncate a file (chop off the end leaving the beginning in place). However, I wouldn't recommend that: this function may not be implemented on some systems. Also, it's much simpler to overwrite the file.