Is there a way to forward (or to get in any other way) the important mail as they arrive (or shortly after that) from the priority inbox to another mail account without forwarding the unimportant mails?I have tried to create a filter with is:important but I get a warning message that the filter will not apply to incoming mail and sure enough the defined actions don't get triggered if an important mail arrives.
Forward only important mail in Gmail
gmail;gmail filters;priority inbox
The only solution that I could find was to access my Gmail account with a 3rd party service like Mail2Web (not free) via IMAP and look for mails in the [Gmail]/Important IMAP folder.I ended up using  Ultimate Notifier because my goal was to get push notifications to my iPhone for important mail. Details on: https://apple.stackexchange.com/q/8250/3566