How much does it cost to connect to private ec2 using aws ssm ?

To achieve this you would need to install the SSM Agent on the EC2 machine and attach an IAM Role to the instance (eg. SSMManagedInstanceCore). There is no additional cost for the agent, you will only pay for the underlying resources (eg. EC2). The instance can be managed using Session Manager and no additional charges apply. You can run command on the instance using Run Command no additional charges apply for this. If you use Systems Manager Automation to automate deployment and configuration using runbooks the pricing is based on the number of steps and duration of execution of steps. The pricing can be seen on this page - https://aws.amazon.com/systems-manager/pricing/