AIR doesn't give you any synchronization mechanisms. It's
entirely up to you how you manage offline operation.
What AIR actually does is provide an out-of-browser runtime
environment for Flex apps, plus some additional APIs that Flex
couldn't give you due to its browser-based nature. Among them are
access to a local database, plus more generalized local filesystem
access. Between these two things, you can store local changes while
offline, then work out some way to push them to the server at a
later time.
You might make your first step into the AIR world just a
simple port of the existing application to run under the AIR
framework. This won't give you any new offline capability, but it
will let the app run as a desktop app, outside a browser. This may
make more sense for your type of application. Then once you get
that going, you can worry about synchronization.
AIR doesn't give you any synchronization mechanisms. It's entirely up to you how you manage offline operation.