Using AWS Fargate as web server
Your website visitors are not supposed to talk directly to the Fargate container. As you realised the IPs are not predictable and can change at any time - it’d be difficult to keep the DNS up to date.
Instead use Application Load Balancer in front of Fargate and use the ALB’s address for your website.
ALB will automatically register the Fargate containers as they come and go.
Hope that helps :)