The only way to fully start or stop your clusters would be to use EBS-backed instances, which the AMI is not based on since they are ephemeral-backed instances.
That being said, if you did move over to EBS-backed instances do choose the optimized instances since the non-optimized instances will cause issues with network contention between Cassandra and EBS usage.
I do have a tool, strictly for dev purposes, that may help. DataStax does not support this tool, but it is used internally and throughout the community.
This is the documentation that may help you:
Just follow the README on the main page and you should get yourself setup and running quite nicely.
After that simply running the s3 store/restore commands grant you what you're asking for, but this has not been thoroughly tested and should only be used in dev environments.
Do take a look at the scripts in case you want to modify anything about them on your own fork:
Hope that helps,