No set up except for the API gateway interface and the lambda function itself, as well as any IAM roles or policies you want to put in place.
No maintenance except for watching for throttling by AWS, watching your billing to ensure it doesn't go out of control, watching for API Gateway errors.
Theoretically, you could also do the entire setup via Cloud Formation; but you can also do the same with EC2 instances and ECS.
What would I do instead? Set up a 1-n member autoscaling group, with rules to scale on load. Set up an ECS service which auto scales on load. Set up an ELB attached to the ECS service. Not as fancy as Lambda and AWS Gateway, but it will probably scale better, at a lower peak cost (at the expense of an up front cost of one server).
No maintenance except for watching for throttling by AWS, watching your billing to ensure it doesn't go out of control, watching for API Gateway errors.