The suggested approach for an architecture running on a container is to have traffic to the container forwarded from a AWS ELB.

If you're using ECS with service discovery then the record being written is actually written to a private hosted zone which is not accessible outside of the VPC(s) it is attached to.

Furthermore containers are ephemeral, with a risk further increased by using a spot instance that the container could be terminated at any point. If the instance is public facing it would need an Elastic IP address to be associated to it so that it could retain the IP address.

You would want to create an A record to this EIP for CloudFront to be able to use it, this A record should have the value of a domain that CloudFront should resolve, and must be in a public hosted zone.

Within your CloudFront distribution the origin value would be set to be the domain name that you setup in the previous step.

You can enable service discovery for Fargate. It uses CloudMap, will automatically create a managed zone and entry inside it. That is the only way to do it without a loadbalancer and hacks. The other way is just to have a lambda function which would update Route53 upon container creation and IP address assignment.