If a plane flies in the opposite direction of the earth's rotation, does this reduce the time for which the plane has to take to get to the destination?
No. Prevailing winds, not Earth's rotation, affect the travel time.
Assume at first that there are no winds. The airplane's engines will accelerate it to the same speed relative to the ground, that's called the "groundspeed", no matter which direction it travels. And to get between one city and another you always have to travel the same distance over the ground. So the travel time, dustance divided by speed, is the same in either direction.
Again, that is with no winds. In the real world there are prevailing winds in the atmosphere where planes fly. Planes flying against these winds take a little longer than planes flying with the wind. In the USA the prevailing winds blow from west to east. That means flying west actually takes a little longer than flying east.