AWS App Runner, when connected to a VPC via a VPC connector, still sends outbound traffic from its own managed ENI in App Runner’s underlying VPC, not through your NAT Gateway. Even though Nat Gateway setup works for Lambda, App Runner does not route traffic through it, so your EIP isn’t the source on the public side.

This is by design, App Runner does not honor the NAT Gateway for outbound.

Reference: https://aws.amazon.com/blogs/containers/deep-dive-on-aws-app-runner-vpc-networking/

Currently, App Runner does not support outbound static IP via NAT Gateway.
Or open a feature request with AWS for adding this functionality.