Faintness isn't all in the head by any means. It's a result of lowered blood pressure which is normal for any fast. Just take things easy and it's usually not too severe. I've done rolling around, throws etc in karate while fasting and dizzy so it's something that we can work with- just take it slowly.
I think it helps to do a light limber up type workout when you wake up. That gives a message to the body that you're not really planning to lie around in bed all day and you actually intend to be active. I loosen all my joints every morning and do some dynamic stretching- which can be done cold- and I think this helps energise my body for the day to come.