Your health and faith are far more important than sports day, so I'd suggest avoiding the run altogether.
It's hard to say whether you'd faint or not, as I don't know how long the fast is or if you're a sporty person, but it's certainly not risking your health over.
Putting It In Writing
Teachers are used to kids making excuses to get out of activities like this, so get a parent to write to your teacher explaining why you've decided not to participate.
You could write the letter yourself, but it seems a bit more credible if you've got your mum or dad to do it.
Explain Clearly
Rather than saying you'd rather not participate, say that you won't be participating. This leaves no room for your teacher to make you do it anyway.
Hopefully, this won't be a problem—it's perfectly reasonable not to run because you're fasting—but you're safer if you make it clear from the outset.
Remember, You've Been Fasting!
Keeping up appearances is important. Even if you've got enough energy to jump and cheer for your team or your friends, don't be too enthusiastic.
If you've got enough energy for that, your teacher might think you're lying about not being able to run. You want to keep their respect, so don't give them any reason to doubt you!