If the Captain's screens are out, does he switch seats with the co-pilot?
If the plane is on final approach and the pilot flying loses his displays the co-pilot (who is also rated to fly the aircraft) may take over. Perhaps more importantly the system is built to avoid this. The "screens" you mention are independent and both (2 screens per side) are capable of displaying the primary instrumentation/information, a situation where both fail simultaneously and completely is rare.  

All screens in the cockpit are generally visible from both seats, any pilot who was a former flight instructor (and many likely were) has plenty of experience viewing instrumentation from the opposing seat. Flying and landing like this should be of little issue.  

In good weather to land an airplane the most important instrument is the airspeed indicator (and perhaps angle of attack indicator if you have one). With that in mind the pilot flying may continue the approach if the backup instruments are still working and in view, that is after all why they are there. 

Some airline operational specs may have specific procedures for this as well but as far as I know the FAA does not really have anything to say on the matter.