It has been possible to run Neo4j in the cloud in a completely automated and managed way for a long time now. In fact, Neo4j provides images to be used in Google cloud or AWS.
Since 2019, Neo4j has also had its own Database-as-a-Service solution with Neo4j Aura, an all-in-one solution used to run a Neo4j database without the need to worry about infrastructure. Check the deployment guide for more information and links: https://neo4j.com/developer/guide-cloud-deployment/.