You can get the last modified time (and a bunch of other stuff) on a remote file using the head() method of LWP::Simple like so...

use LWP::Simple;

my $url = "http://www.perlmonks.org/yourfile";

my ($content_type, $document_length, $modified_time, $expires, $server
+) = head($url);
[download]

Docs for LWP::Simple are here