If the volume of two containers stays the same , and one containers temperature is lowered, what happens to the pressure in the cooler container ?

Gay-Lussacs law states that pressure is directly proportional to temperature at constant volume.
in this case the volumes of the 2 containers remain the same therefore volume is constant.
P/ T = k
where P - pressure, T - temperature and k - constant 
when temperature decreases the pressure too decreases and vice versa.
if temperature of one container is lowered the pressure also reduces.
since one containers temperature is lowered the pressure in that container reduces. 

answer is when temperature is lowered, pressure too is lowered.