How to find out the time spent by a specific user on a case record from its 'New' to 'Closed' status?

Well since spring 21 still isn't fully out otherwise could have achieve the below with that but with trigger here it is

• First you can create an custom SObject Case Workflow with the following fields

You can use the following trigger, please note this is just to give you an idea. You can change the trigger as per your need

trigger CaseStatusWorkflowTrigger on Case (after insert, after update) {
	List<Case_Workflow__c> cwToInsertList = new List<Case_Workflow__c>();    
    for(Case caseRec : trigger.new){
        if(trigger.oldMap == null || (caseRec.Status != trigger.oldMap.get(caseRec.Id).Status))
        {
            Case_Workflow__c cwToInsert = new Case_Workflow__c();
            cwToInsert.OldValue__c = trigger.oldMap?.get(caseRec.Id).Status;
            cwToInsert.NewValue__c = caseRec.Status;
            cwToInsert.FieldName__c = 'Status';
            cwToInsert.Case__c = caseRec.Id;
            cwToInsertList.add(cwToInsert);
        }
    }
    if(!cwToInsertList.isEmpty())
        Database.insert(cwToInsertList);
}

Test class

@isTest
public class CaseStatusWorkflowTest {
    private static testMethod void testCaseStatusWorkflowOnInsert() {
        Test.startTest();
        Case caseRec = new Case(Status = 'Open', Origin = 'Phone');
        insert caseRec;
        Test.stopTest();
        Case_Workflow__c cw = [SELECT Id, OldValue__c, NewValue__c FROM Case_Workflow__c WHERE Case__c = :caseRec.Id];
        system.assert(cw.OldValue__c == null);
        system.assert(cw.NewValue__c == 'Open');
    }
}