Bitnami provides ready-to-use images in different cloud providers and you will only pay the cloud provider for the usage of the resources. That means that you won't pay anything to Bitnami or to Neo4j for its usage but you need to pay for the usage of the infrastructure in the cloud.
As you mentioned, AWS has a free tier that allows you launch micro instances.
However, Neo4j requires a higher instance to run.
Let us know if you have any questions.