Try:
LVContacts Sync for Zimbra and Google Android™
LVContacts currently syncs the following items between Zimbra (including OpenSource) and Android phones: Contacts (multiple address books), and shortly (in November theoretically) (multiple) Calenders and Tasks. Eventually they are aiming for email folders. In the meantime, you could you Imap or Pop, and use LVContacts for tasks, events, and contacts. (It doesn't use ActiveSync as the sync protocol).