For the redirection part, it's a poor idea to try and call the same module using both method and function calling. One or the other really needs to be chosen.

Option 1: Make it all function calls by importing redirect, as well.

#!/usr/bin/perl -w
#use strict;
use Archive::Zip qw(:ERROR_CODES :CONSTANTS);
use CGI qw(:standard :nph redirect);

my $id;
if ($id = param('link')){
   my $obj = Archive::Zip->new();
   my $file_member = $obj->addFile( $id );
   if ($obj->writeToFileNamed('download.zip') != AZ_OK) {  
     print header(type => 'text/plain');
     print "Bad something or other!";
   } else {
     print redirect('http://domain.com/files/download.zip');
   }
}
[download]

Option 2: Make it all method calls, by not importing any functions.

#!/usr/bin/perl -w
#use strict;
use Archive::Zip qw(:ERROR_CODES :CONSTANTS);
use CGI qw(:nph);

my $q = CGI->new();
my $id;
if ($id = $q->param('link')){
   my $obj = Archive::Zip->new();
   my $file_member = $obj->addFile( $id );
   if ($obj->writeToFileNamed('download.zip') != AZ_OK) {  
     print $q->header(type => 'text/plain');
     print "Bad something or other!";
   } else {
     print $q->redirect('http://domain.com/files/download.zip');
   }
}
[download]

I've also done three other things:

1. I made it a non-parsed-headers (nph) script. Otherwise, the first thing that is send by Apache (or other web servers) to your browser is a 200 OK header, as far as I can recall, and the redirect won't work. (When I write a script for my web server, I make it nph if I do anything other than output standard pages, so I'm not sure.) You'll need to tell your web server that it's a nph script. [For Apache, you just need to call it something that begins with 'nph-', I don't know about others.]
2. I put the $id assignment in the if statement. That should work (I haven't -c checked this code) and it eliminates having to call CGI->param('link') twice.
3. I assumed you were sending text to the browser, rather than HTML, for the error checking, since it didn't look like HTML. If it is, take out the type parameter.