There is a lab on Serverless Architecture in the Architecting course, making use of S3, Lambda, SNS, SQS and DynamoDB to process a file of customer banking transactions.
However there is no lab on API Gateway. Even though hands-on knowledge is not required for the exam, it is always good to try things out.
There is a walkthough Build a Serverless Web Application on the AWS site.
The exercise uses API Gateway, S3, Lambda and DynamoDB. For good measure, although it is not covered in the Architecting course or exam, it also uses Cognito for authentication.
The exercise takes about 2 hours. Cloud Formation scripts are also provided, but then you miss all the learning points.
Basically you sign in to the application using your email address, and then click on a map to order a taxi, except in this case its a flying taxi in the form of a Unicorn. When you click on a map and order a Unicorn, your coordinates and username are posted to API gateway, which triggers Lambda to find the nearest available Unicorn. Lambda also logs your ride in DynamoDB.