most of my blue prince holly’s leaves have turned brown and brittle. Part of the shrub is alive with healthy green leaves should
I cut the shrub back? Should I cut the branches back as most have green leaves and the holly seems to still be alive?
It would appear that you are just outside of their native region. Without winter protection during the colder parts of the year, it can kill off much of the growth. Sometimes it will be killed all the way back to the ground. I would make sure to give plenty of protection in winter, such as mulching, and wrapping the tree in burlap during the night.
You can safely cut out any dead growth. This will help promote new growth.