How can I synchronise Always Encrypted data from On-Premises to Azure SQL Database (near real time)?
In general, any tool that copies data (e.g. using SELECT/INSERT or bulk select/copy) should work. If the target database (in SQL DB) uses a different set of keys than the original database (in SQL Server), the tool needs to have access to both keys (and it would need to re-encrypt the data during the migration). If the target database using the same keys as the source database, it should be possible to avoid re-encrypting the data (please, see https://blogs.msdn.microsoft.com/sqlsecurity/2016/01/07/best-practices-for-moving-data-encrypted-with-always-encrypted/). 

SSIS support both options to migrate the data (with and without re-encryption), but I'm not sure if it meets your near-real time requirement. You will basically need to run the SSIS job periodically. 

Stretch Database supports moving encrypted data (without re-encrypting), but does not replicate encryption metadata. To retrieve the plaintext values or run queries on the part of the table that "stretches" to SQL DB, an application needs to connect to SQL Server (which has encryption metadata and will delegate the query to SQL DB). Your app can't access plaintext by connecting to SQL DB directly.