Your hero can simply use a plumb bob.
Until the modern age, plumb-bobs were used on most tall structures to provide vertical datum lines for the building measurements. A section of the scaffolding would hold a plumb line, which was centered over a datum mark on the floor. As the building proceeded upward, the plumb line would also be taken higher, still centered on the datum.
During the descent to the center and during the climb on the other side, thanks to the shell theorem your hero won't have to care about the mass of Earth above him. The plumb bob will always point to the center of mass of the part of the planet beneath their feet.
Once they reach the center, they simply have to turn it around.
To know when the center has been reached, it is sufficient to hang a mass to a spring of known elastic constant. Since the weight of the mass will decrease until reaching zero at the center, the deformation of the mass is a good indication of the reached depth.