What is the best way to Replicate some tables from on premises database to Azure SQL Cloud without using Microsoft DataSync feature?

You can use SQL Server Mgmt Studio (SSMS) and take a backup and then restore the backup to an already created azure sql DB. You can also look at this MVA course by Microsoft to migrating existing on-prem SQL DBs to Azure SQL, https://mva.microsoft.com/en-US/training-courses/migrating-sql-server-databases-to-azure-16594

Backup and restore process (SQL Management Studio ) will be the another best way to replicate table from onpremise to azure