set up a postgres database on a cloud vm, install docker to it, and link in the image

Beware the 0x repository your mention is listed as deprecated and unsupported, so I would not trust or use that container in production systems. If you just want a proof of concept, it likely will work fine. When it comes to running containers, it really depends on your needs, and there are several options:

• You can always spin anything up you want on an Amazon Elastic Compute Cloud (EC2) Instance and manage Docker 100% yourself.
• Using Amazon Elastic Kubernetes Service (EKS) you can create tasks to run the container. (I don't recommend this for your first time with containers)
• Using Elastic Container Services(ECS) you can create tasks (where you specify all the required ENV Variables)here and run them on AWS Fargate and are completely serverless. This is a great way to run a container, especially when you are researching options.
• Using Amazon RDS you can spin up a dedicated Postgres instance that is managed and patched by AWS. This however will not run your API, but it will give you a stable and well-maintained postgres instance.
• You could run postgres as a container if desired, but I would not recommend it.

As for sizing, I always start with small sizes first and test to see if I need to vertically scale the hardware this keeps costs as low as possible . With AWS you can vertically scale an instance quite easily. Using Fargate with ECS you can also specify the size of the Task, and I typically do the same things.