You don't state which version of Zimbra you have, but you would need to purchase the network edition and either sync it off your Outlook and/or purchase the mobile option. There are no real easy ways to do this with the free version. You can look at the Funambol option, but it's going to take some work on your part.
Take a look at this link to see which option you can and can't use with the various versions.
Zimbra - Product Editions
Release 6.0.2_GA_1912.UBUNTU8_64 UBUNTU8_64 NETWORK edition + Mobile Option
Activesync with Moto Q9C, HTC Touch Pro, Palm Pro, & Palm Pre