Solution:

Let the expected number of minutes it will take the rook to reach the upper right corner from the top or right edges be $E_{e}$, and let the expected number of minutes it will take the rook to reach the upper right corner from any other square be $E_{c}$. Note that this is justified because the expected time from any square on the top or right edges is the same, as is the expected time from any other square (this is because swapping any two rows or columns doesn't affect the movement of the rook). This gives us two linear equations:
$$\begin{gathered} E_{c} = \frac{2}{14}\left(E_{e} + 1\right) + \frac{12}{14}\left(E_{c} + 1\right) \\ E_{e} = \frac{1}{14}(1) + \frac{6}{14}\left(E_{e} + 1\right) + \frac{7}{14}\left(E_{c} + 1\right) \end{gathered}$$
which gives the solution $E_{e} = 63$, $E_{c} = 70$.