Sounds like an exercise-induced migraine/headache to me which is often caused by one of two things.
Dehydration - make sure that you drink well during fast day and in the morning.
Too high a heart-rate; in spin, when they tell you to take it to 80 or 90% intensity, that should fall in with your max heart rate. If you go too far beyond that for long, you will end up with headaches - and you're not really going to improve your fitness by over-stressing the heart. For example, if you're told to do 2 minutes at 70%, then a minute at 80% and then 30 seconds at 90% before dropping back down to 80, it's interval training that will work your heart. If, instead, your 70% is really 80% and your 80 is 90% and your 90 is 100, then you're potentially overtaxing things and won't see the benefits - and end up with headaches.
I used to get similar headaches and my spin instructor buddy (v experienced and 'mean') told me that it's a warning sign that something's not right; particularly if they hang around for a few hours. He encouraged me to get an HRM and keep an eye on it during class - and it turned out that I was pushing it too hard. A few weeks of being more careful (which felt v odd at times since I didn't feel that I was working hard enough at the lower intensities) and my fitness levels improved so that my HR was now lower at higher resistance/rpms and I could work back up again. Just a thought!
Having said all that, if you don't mind the headache, nothing physical should stop you spinning after a fast day. Don't go overboard on the breakfast unless it's something easily digestable that will send glycogen straight to the muscle. If you eat at 7 and then do spin at 9, you don't want your body still be working hard to digest stuff with the blood flow in the digestive system rather than your muscles.