Your major problem is that you are trying to interpolate a function call (cookie("host")) into a double-quoted string and Perl doesn't do that.

However, you are making the logic a lot more complex than it needs to be. I'd do something like this (untested):

my $host = cookie('host');
unless ($db->do("update logs set visits=visits+1 where host='$host')" 
+> 1) {
  $db->do("insert into logs values (null, '$host', '$xronos', '$file',
+ 1)");
}
[download]

I've omitted error checking for clarity.