When connecting to a SQL Server Cloud SQL instance from Cloud Run,
If connecting via Public IP - make use of Cloud SQL connectors
If connecting via Private IP - configure Cloud Run with VPC access (Direct VPC egress or a connector) on the same VPC as your SQL instance
Reference - https://cloud.google.com/sql/docs/sqlserver/connect-run#connect
Since the application is a .NET application, an alternative approach that would help connecting to a SQL Server instance would be using Cloud SQL Auth Proxy as a sidecar container in your Cloud Run service.
Here's a guide on how to - https://cloud.google.com/blog/products/serverless/cloud-run-now-supports-multi-container-deployments