Nursery school not giving application because she is 1 month too young... is it fair?

Most nursery schools set their admission date to conform to the local school district's admission date. That way, children learn with the same group they will be with throughout their school years. Also remember that readiness has as much to do with social and emotional readiness as it does with academic skills. You well may find her going to school with kids who are both more and less capable than her. That's what life is about. If you push her ahead now she will simply have to repeat a year as her friends all go off to kindergarten. I think the school is taking the right position.