AWS Cloud-Shell - Connection on Private IP

No, this is not yet available, below are relevent questions and answers from the CloudShell FAQ around this topic

Q: Can I access resources in a VPC with CloudShell?

No, you cannot currently access resources that are in your private VPC in this release of CloudShell.

Q: What is AWS CloudShell?

AWS CloudShell is a browser-based shell that makes it easier to securely manage, explore, and interact with your AWS resources. CloudShell is pre-authenticated with your console credentials. Common development and operations tools are pre-installed, so there’s no need to install or configure software on your local machine. With CloudShell, you can quickly run scripts with the AWS Command Line Interface (AWS CLI), experiment with AWS service APIs using the AWS SDKs, or use a range of other tools to be more productive.

Q: What can I do with CloudShell?

CloudShell gets you started with the AWS CLI more quickly, so you can automate tasks, manage infrastructure, and interact with AWS services. You can use CloudShell to clone repositories containing commonly used scripts, make edits to those scripts, and store them for future use. You can use AWS SDKs to develop applications and use common CLIs, such as the Amazon Elastic Container Service (Amazon ECS) CLI and the AWS Serverless Application Model (AWS SAM) CLI, to manage your AWS resources. You can save your work at no cost in 1 GB of persistent storage available in your home directory.