I don't have a barn manager since I don't board my horses but here's my thoughts on what one should be.
1. Knowledgeable about horses and their care able to spot a lame or ill horse and know what needs to be done whether treating it themselves or calling a vet in the event the owner cannot get there
2. Business minded but not so much that all they see is dollars & cents.
3. Communication communication communication
4. Good listening skills and open mindedness not bent on their way is the only way things should be done.
5. Caring about the horses genuine well being
6. Attentiveness-to what goes on and around the barn
7. Good sense of humor no one likes working with a cranky grouch
8. Patience for dealing with the horses and their owners
9. Legal knowledge of barn management
10. Excellent book keeping/record keeping skills
That's just a few of what I feel is important