Mesh current method question here?Is this correct?
The first two equations you wrote are correct. You made mistake while re-arranging them. The equations should be:

$$5 + V_i = 2k\times I_1 - 1k\times I_2\tag1$$
$$5 = 1k\times I_1 - 2k\times I_2 \tag2$$

Solving which you will get the value for \$I_2\$. 
$$I_2 = \frac{-5+V_i}{3k}$$

And then,
$$V_o = I_2 \times 1k = (V_i-5)/3$$

It was more easy if nodal analysis or superposition theorem was used.