public virtual void ApplyStyle( 
   StyleItem styleItem
)

{
  //Note: Must be on QueuedTask.Run
  //Get the Style project items in the project
  var styleProjectItems = Project.Current?.GetItems<StyleProjectItem>();
  //Get the ArcGIS 2D Style Project Item
  var styleProjectItem =
  styleProjectItems.FirstOrDefault(s => s.Name == "ArcGIS 2D");
  if (styleProjectItem == null) return;
  //Get the north arrow style item you need
  var northArrowStyleItem =
  styleProjectItem.SearchSymbols(StyleItemType.NorthArrow, "ArcGIS North 18").FirstOrDefault();
  if (northArrowStyleItem == null) return;
  //Select a North arrow layout element
  var northArrowElement = layout.GetSelectedElements().OfType<NorthArrow>().FirstOrDefault();
  if (northArrowElement != null)
  {
    //Check if the input style can be applied to the element
    if (northArrowElement.CanApplyStyle(northArrowStyleItem))
      //Apply the style
      northArrowElement.ApplyStyle(northArrowStyleItem);
  }
}