CloudBlockBlob doesn't have any method to set the blob tier to hot/cool/archive. I have also checked the other blob types and they do not have a method that allows this either.
I.E this method: https://learn.microsoft.com/en-us/rest/api/storageservices/set-blob-tier
Is their any way to change the blob tier in code from hot to cold in C# with azure storage?
I think the method is exactly what you need: CloudBlockBlob.SetStandardBlobTier. Maybe you were not checking the latest version of Azure Storage Client Library?