How will the current flow in the figure drawn below? 
Consider the green material in the figure to be conducting. So, I was wondering how the current will flow in the rod, as the battery is not connected at the ends, but on the surface of the rod. Thus, will there be no current at the ends of the rod and what about voltage?
 A: Yes. There won't be any noticeable amount current at the ends of the rod because the electric field formed inside the conductor will be restricted so the electrons will dominantly experience the force when they are in the region between the points across which the potential difference is applied or else there might be some small amount influence of electric field at the end and the area near the end, which will result in some negligible amount of current.
And the same thing goes with potential difference. The potential at one end and the point at which the wire is connected will be nearly same and hence the potential difference will be nearly equal to zero and you won't find any significant amount of current behind those two points at which the wires are connected.
While doing some ideal calculations you can also neglect the resistance of that region of the rod which is on the left of positive terminal and same with which is on the right of negative terminal to get some near to accurate results.
A: An approximate numerically calculated figure is attached.
The figure is a two-dimensional result.
Although 3D calculations is possible, the figure is easier to see in 2D.
$\phi$ is potential in volts.
$\vec{J}$ is the electric current density.

(Edit #1)
I add a figure highlighting the elements in the two left-most columns.

