Start a timer after instantiating a game object

I'm working on a variant of the Space Shooter game from Unity's tutorial. I have created a power-up object that instantiates a shield for your ship when touched, and I'm trying to implement a bar that indicates how long the shield will last. The bar itself is working fine, but I can't get the bar to start depleting the moment that the shield is instantiated.

My code for the GUI and shield looks like this:

     private bool pup = false;
     public float barProgress;
     public Vector2 pos = new Vector2(10, 50);
     public Vector2 size = new Vector2(20, 250);
     public Texture2D emptyTex;
     public Texture2D fullTex;
 
     void OnGUI()
     {
         if (pup)
         {
             GUI.BeginGroup(new Rect(pos.x, pos.y, size.x, size.y));
             GUI.Box(new Rect(0, 0, size.x, size.y), fullTex);
 
             GUI.BeginGroup(new Rect(0, 0, size.x, size.y * barProgress));
             GUI.Box(new Rect(0, 0, size.x, size.y), emptyTex);
             GUI.EndGroup();
             GUI.EndGroup();
         }
     }
 
     void OnTriggerEnter(Collider other)
     {
         if (other.tag == "pup")
         {
             GameObject go = Instantiate(shield, shieldSpawn.position, shieldSpawn.rotation) as GameObject;
             go.transform.parent = GameObject.Find("Power Ups").transform;
             audio_shieldUp.Play();
             audio_shieldDown.PlayDelayed(shieldDuration);
             Destroy(GameObject.FindWithTag("Shield"), shieldDuration);
             Destroy(other.gameObject);
             pup = true;
         }
     }
 
     void Update()
     {
         if (pup)
         {
             barProgress = Time.time / shieldDuration;
         }
     }

The shield instantiates and the bar shows up on screen, but the "barProgress" counter has already started counting and depletes too soon. What am I doing wrong here?

Also, any suggestions for how to remove the bar after shieldDuration has ended, as it's drawn in the script and not instantiated?