Outlook 2010 and ActiveSync (Z-Push)

These are both possible. For Outlook 2007, the process is called Outlook Anywhere and for Windows Mobile, it is called Activesync & Direct Push. Both of these are available "out of the box" with Exchange 2007 but do require a little configuration on the server side (Exchange 2003 SP2 also provides these capabilities).

For Outlook, if you do not have the autodiscover service working, then you need to manually configure Outlook. To do this go to Tools > Account settings and double-click on your account name. Click the More Settings button, go to the Connection and check the box under Outlook Anywhere.

Then select the Exchange Proxy Settings box and in the URL type in the address for your OWA server. Check the two SSL boxes assuming SSL is configured correctly on OWA. Then turn on the setting for HTTP on slow networks. OK your way out of the menus and you should be set (Outlook may require a restart).

For Activesync, simply enter in the address of your OWA server, enter your authentication information, and enable SSL.

If you need help with the server-side, then we will need to know what version of Exchange you are running.

ad1) Please be more specific with sync issues. Do messages disappear? Do messages get duplicated?...

ad2) The basic issue is that Outlook 2003 support UDP and polling notifications. Exchange 2007 supports UDP, polling and Async notifications. Exchange 2010 only supports polling and Async notifications. This means when Outlook 2003 move from Exchange 2007 to Exchange 2010, Outlook clients will fall back to polling which by default only gets notifications every 30secs-1min. This means any change won’t show up immediately.

UDP notification support was removed from Exchange 2010. As a result, Outlook 2003 can only use polling notifications in online mode, which are still supported by RPC Client Access. This will result in a slight delay in updates to item status (30 seconds on average up to a 1 minute delay) when changes are made to items in a mailbox accessed by Outlook 2003.

source: http://social.technet.microsoft.com/Forums/en/exchange2010/thread/d5a610d5-b3cd-40a5-8653-634f3b174f4e