Your requirement is valid and amazon knows:

When you stop an instance, we shut it down. We don't charge usage for a stopped instance, or data transfer fees, but we do charge for the storage for any Amazon EBS volumes.

Reference :
- https://docs.aws.amazon.com/AWSEC2/latest/UserGuide/Stop_Start.html

Here some approaches to start/stop your instaces

Amazon EC2 HTTP API

This is an api rest and you can perform a simple http request to start or stop your instance:

• Amazon EC2 API Reference
• Start Instance endpoint

https://ec2.amazonaws.com/?Action=StartInstances&...

• Stop Instance endpoint

https://ec2.amazonaws.com/?Action=StopInstances&...

You can invoke this api from Jenkins in many ways : simple shell execution,groovy and scripted/declarative, pipelines.

AWS CLI

• start instance
• stop instance

Here more about how suspend instances using aws cli:

• https://docs.aws.amazon.com/autoscaling/ec2/userguide/as-suspend-resume-processes.html
• Also with powershell: https://docs.aws.amazon.com/AWSEC2/latest/UserGuide/Stop_Start.html

You can invoke this api from Jenkins in many ways : simple shell execution,groovy and scripted/declarative, pipelines.

AWS Instance Scheduler

In 2018, AWS launched the AWS Instance Scheduler, a new and improved scheduling solution that enables customers to schedule Amazon EC2 instances.

With this tool you can automatically start and stop the Amazon EC2 and Amazon RDS instances.

Reference :

• https://aws.amazon.com/answers/infrastructure-management/instance-scheduler/

With this approach you don't need Jenkins :b