I sprained my arm a little over a week ago. How long until I can start my boxing/kickboxing class again?

Common sense. Start your boxing/kickboxing slowly- striking can reinjure your arm and worsen your symptoms. If the ligaments are strained, you could do significant damage by doing too much. Keep your strikes to a minimum until the arm is pain free. If you can air-box without pain, try that activity for a couple of weeks. Your family dr. Can help guide you further. Good luck.