I wouldn't use DataSynch because it creates many objects in your database to perform synchronization (it's an invasive solution). You can indeed try the Import/Export feature; that should work fine. You can also download a trial version of the Enzo backup tool, which comes with a 30-day free trial: http://www.bluesyntax.net/backup.aspx. [disclaimer: I am the author of this tool]
Regarding the pricing question, you may be charged for data being extracted out of the database. Moving data "in" SQL Azure is free of charge for now. If you are transferring the data to a different data center, you will be charged for extracting the data. It's 15 cents per GB in the US and Europe, and 20 cents in Asia. Here are the pricing details: http://www.microsoft.com/windowsazure/pricing/
Keep in mind that a database that requires 4GB of storage doesn't mean you have 4GB of data. Sometimes indexes can take a lot of space. To estimate the size of the data you will need to transfer you can either drop your indexes (and wait a little for the database size to shrink; the database size should be roughly equal to your data transfer needs) or you can calculate the size of your tables by running a command. Here is a link to an article that shows how to do something similar (look at the second command with is a SELECT statement; just run it for all the tables): http://www.sqldocumentor.com/table-size-in-sql-server-find-rows-and-disk-space-usage