This is answered in the novel (emphasis mine):
I have made the effort, and God helping me, have come safely back to this room. I must put down every detail in order. I went whilst my courage was fresh straight to the window on the south side, and at once got outside on this side. The stones are big and roughly cut, and the mortar has by process of time been washed away between them.
Dracula, Chapter 4
So Harker is emphatically not using supernatural powers. The walls of medieval castle are not as smooth as modern brick walls, especially not when they're as old as Dracula's castle, and there was enough deterioration for Harker to keep a tenuous grip.