Yes, the dead-looking part is dead and can't be revived. If you cut all of the dead stuff down to the ground, this could stimulate the rose to produce new basal shoots from the bottom. Keep the green canes and get rid of the brittle brown ones.
Now that I've seen the cluster of blooms in your latest photo, I've changed my mind: This could be Aloha, a 1949 climbing hybrid tea that's still popular today. It's a short climber that usually looks more like a shrub, growing to about 5-6 feet tall. It's hardy to zone 5, so it easily could have been growing in your garden for decades.
