Is it possible to directly access MySql databases through an Adobe AIR/Flex
application?
If not, what is the next best alternative?
No, although AIR does have a local (client-side) SQLite database builtin. You don't really want your client apps to have direct access to the server-side database anyways, it can really cause weird problems and open security issues up. You can expose a MySQL database with web services however. This article has a really easy way of doing it.