To manually test a GraphQL API using Postman, start by cloning the Apollo Server Starter project and installing the necessary dependencies. Review the schema of your API to understand how data is structured. Then, open Postman and create a new query with the URL of the request, selecting POST as the method. Formulate the body of the request by choosing GraphQL from the Body options, then write or paste your query into the Query window. Finally, add variables to apply sorting and filtering if needed, and click the Send button to execute the query. This allows you to verify that your API is working correctly, and can be used in conjunction with a GraphQL IDE like Apollo Explorer for more advanced testing and development capabilities.