If my Jolteon was to use Secret Power in grass, it turns into Needle Arm, a grass type move. However does this mean that the category of the move would be shifted from Physical to Special?
Unfortunately, Le Scaf's answer was incorrect. Secret Power only has variable animation and secondary effect. It is always Normal type unless the user's ability is Refrigerate, Aerialate, or Pixilate. Therefore, it is always physical in Gen 3.
Secret Power fully takes on the move in terms of category.