I am a female, at work I use an electric drill gun, for about to months my fingers are always so sore should I use ice or heat?

Ice after heat befor. Often it is better to use ice after you've done something to aggrevate your hand. Heat is helpful for the aching & stiffness that may persists at night or into the next day. You may also consider using a padded glove, like a biking glove, or exercise glove, to decrease the impact on your hand.
Both plus Advil (ibuprofen) Typically a warm heat helps loosen things up before activity and ice/cold helps after. I think lots of folks don't really like cold on their fingers, though. I would suggest trying to 2 advil (ibuprofen) after and see how that works.