I have been told that this is possible but I don't quite understand how to implement that with getopt::std.

That's because you should use Getopt::Long. Also

open STDOUT, ">> $log_dir/$log_file
[download]

Don't open files like this if you don't know what Perl does with two-argument open. Especially if $log_dir and $log_file come from user input. Do it like this:

open STDOUT, '>>', $log_dir/$log_file
[download]

Consider using File::Spec::Functions for manupilating file names.

&filesize;
[download]

Don't call functions like this if you don't know what it actually does. Call them like this:

filesize();
[download]

use warnings; instead of -w. Also use strict;