I want to host a neo4j graph database somewhere to i can create an api for it. Where do i host something like this? Their proprietary hosing called AuraDB is a little expensive, even for the base option.
AuraDB has a free tier so you can use it for development. Once your API is working then you can host it in AWS or Google Cloud or AuraDB.  You will get what you paid for.
How to host Neo4j Graph db