Final answer:
The most cost-effective method for mounting a network file system for usage that decreases over time is to employ a service like AWS Elastic File System with an Infrequent Access storage class, providing lower-cost storage for less frequently accessed files.
Step-by-step explanation:
The most cost-effective solution for mounting a network file system on Linux instances that will be used frequently at first and then infrequently would likely involve a service that offers a combination of performance and long-term storage pricing.
An example of such a service is AWS Elastic File System (EFS) with its Infrequent Access (IA) storage class. Using EFS, you can store files that are accessed less frequently in a lower-cost storage class while maintaining the same file system access semantics. The EFS IA storage class costs less than the standard storage class and is designed for files accessed less regularly, thereby offering cost savings as your access patterns change over time.