Your criteria is simply host = $host. Your object should set timestamp to the contents of @new. Your options include multiple which tells it to update all matching records and safe which causes it to croak on errors (so the || die you have is superfluous).
Are you getting any errors or do the records just end up empty?
Based on the docs, it looks right. So what's missing? We don't know. Without seeing more of the code, I don't think anyone will be able to help you.
I'm hoping that I can learn a bit about MongoDB from trying to help you. Is the update happening within the @cols foreach loop? How are you getting $col? $col = $db->$cols;?
Based on your @new foreach print, it looks like you are storing multiple timestamps in each record. Is that what you intended? If so, it seems odd to me to store timestamps in a column named timestamp. :)