Judging by they type of pain you may have, it may be because you don't move your neck around a lot. Try stretching. It will hurt, but it will ease and relax your muscles. If the pain is too much, then start with very little stretches.
Put a heating pad on your neck. It will soothe your muscles too. Ice also has the same effect. Remember to keep moving it. Avoid any sudden movements. That could make your neck worse.
You could also use pain relievers. Over-the-counter medicines include ibuprofen and acetaminophen. Massage the area. Do it very lightly.
Turn your head the opposite direction of your pain. Since it hurts on the right side, turn it towards the left side.
I hope your neck starts feeling better.
Other than that, I hope your babysitting goes well! :)