Answer
It takes the sun $127.6s$ to moves a distance equal to its own diameter.
Work Step by Step
Since we conceive that the sun revolves around us, its movement follows a circle of radius $r$, which is the distance between the person and the sun. Therefore, the diameter of the sun $s$ is an arc defined by angle $\theta_{sun}$ $$s=r\theta_{sun}$$
Take a look at figure 8.5 in the book for better understanding.
This means the sun moving a distance of its diameter means the sun is moving at an angular displacement $\Delta\theta=\theta_{sun}=9.28\times10^{-3}rad$
The sun's revolving speed is the speed the earth spins on its axis, which means $\overline\omega=2\pi rad/24h=7.27\times10^{-5}rad/s$
Therefore, $$\Delta t=\frac{\Delta \theta}{\overline\omega}=127.6s$$
