SQL Server 2012 import/export wizard not showing at all
The import/export wizard is not showing because it is not available for Azure SQL. Even if you downloaded the latest version of SSMS, you shouldn't see it. 

You are in a different environment that how on-premise SQL Server databases work. In that Azure SQL does not support using cross database queries. So doing an insert into or select insert are not going to work. 

Depending on how big the table is, your easiest thing to do is using the script object wizard and just include the data. Then just run that under the destination database.

If you need to do this on a routine basis then look at the data sync services Azure offers.