Answer
The asteroid would have had to arrive $~~8.1~hours~~$ later.
Work Step by Step
We can find the angle $\theta$ between $102^{\circ}~E$ and $20^{\circ}~W$:
$\theta = 102^{\circ}+20^{\circ} = 122^{\circ}$
We can find the time it takes the Earth to rotate through this angle:
$t = (\frac{122^{\circ}}{360^{\circ}})~(24~h) = 8.1~h$
The asteroid would have had to arrive $~~8.1~hours~~$ later.
