@isTest
public class CaseUpdateEmailHandler_Test {
    
      static testMethod void testEmailHandler(){
        
        //Create a dummy account and other records that your case object need. 
        //I recommend using a smartfactory or test data utility to create test data.
     	Account testAccount = new Account(name='Test Company Name');
     	insert testAccount;
  
        //Create a dummy case. Fill the fields you need
	 	Case ca = new Case(Subject='Test Case Update Email Handler');
        ca.AccountId = testAccount.Id;
		insert ca;
        
        // create a new email and envelope object
        Messaging.InboundEmail email = new Messaging.InboundEmail() ;
        Messaging.InboundEnvelope env = new Messaging.InboundEnvelope();
        
        // setup the data for the email
        email.subject = 'Test email handler:' + ca.Id; //Since you are looking for case Id in the subject, add Id of the case you created
        email.fromname = 'FirstName LastName';
        email.plainTextBody = 'Status:New' +  '\n' + 'Priority:High';
        env.fromAddress = 'someaddress@email.com';
  
        CaseUpdateEmailHandler emailProcess = new CaseUpdateEmailHandler();
        Messaging.InboundEmailResult result = emailProcess.handleInboundEmail(email, env);
        
        //Assert results
		
        //Test for various cases like what happens if there is no email body,no case Id in the subject etc
    }
}