They will understand about the fainting, they see this kind of thing all the time, they will only be interested in making sure your arm is okay and maybe checking blood sugars/blood pressure to see why you fainted.

What you have written sounds fine by me, you could ask if you could see someone as soon as possible, the worst that can happen is they'll possibly say no and give you a later appointment. Remember this is all very routine for them, they won't be fazed by it.

(((hugs)))

Let us know how it goes.