I remember Time Triling along that stretch of road back in the 80sRiding from Durham to Chester le Street the path runs on the East side of the dual carriageway so you're facing the oncoming traffic which means you don't have to give way to traffic coming from behind you at the T junctions. The Arnison Centre roundabout is a PITA whichever way you approach. The biggest problem though is getting back across the A167 to get into C-l-S. I prefer to ride on the road the whole way to avoid this.
The path is just a shared pavement at the side of the main road. It's been resurfaced in the last couple of years and is still in fairly good nick. Think the '50s style path you're thinking about is between Nevilles Cross and Cock o' the North so heading South out of Durham.