While it is possible, I don't think this is a spider bite. Most of the time a spider will only bite when disturbed and I would think it would be the nose or a paw. I am on a Facebook group that has helped me understand spiders and their behavior a bit more. I still am creeped out by them, but I am trying to get past it a bit because they are beneficial.
https://www.facebook.com/groups/antmans.hill
Of course it can't hurt to put a compress on it.