To derive this you'll want to use the Frisch-Waugh-Lovell theorem.
Using the true variable, $x_2$, let $\widetilde{x_2}$ be the residual from a regression of $x_2$ on $x_1$,
$$x_2 = \delta_0 +\delta_1 x_1 +\widetilde{x_2}$$
We thus have,
$$\bar{x_2} = \delta_0 +\delta_1 x_1 +e +\widetilde{x_2}$$
The residual from a regression of $\bar{x_2}$ on $x_1$ is $(e +\widetilde{x_2})$.
By the Frisch-Waugh-Lovell theorem, the OLS estimate of the coefficient for $x_2$ in your model of estimation will be the same as the OLS estimate from
$$y_i = \alpha_0 +\alpha_2 (e_i +\widetilde{x_{2i}}) + u_i$$
So we have $$\hat{\alpha_2} = \frac{Cov(y_i, (e_i +\widetilde{x_{2i}}))}{Var(e_i +\widetilde{x_{2i}})}$$
You will plug in for $y_i = \alpha_0 +\alpha_{1i} x_1 +\alpha_2 x_{2i}+u_i$, and note that $Cov(x_{1i}, \widetilde{x_{2i}})=0$ because $\widetilde{x_{2i}}$ is the residual from an OLS regression with $x_1$ as a regressor. To proceed, you will need to make an assumption regarding $Cov(x_{1i}, e_i))$ and $Cov(u_{i}, e_i))$.
To estimate the effect of $x_1$ on $y$, we need to consider a regression of $x_1$ on $x_2$.
$$x_1 = a_0 +a_1 x_2 + \widetilde{x_1}$$
Using the mismeasured version of $x_2$,
$$x_1 = a_0 +a_1 \bar{x_2} - a_1e + \widetilde{x_1} $$
The residual is $(- a_1e + \widetilde{x_1})$.
We apply the Frisch-Waugh-Lovell theorem to know the estimate for the coefficient of $x_1$ is the same as the estimate from,
$$y_i = \alpha_0 +\alpha_1 (- a_1e + \widetilde{x_1}) + u_i$$
This is $$\hat{\alpha_1} = \frac{Cov(y_i, (- a_1e + \widetilde{x_1}))}{Var(- a_1e + \widetilde{x_1})}$$
Analogous to before, you will plug in for $y_i = \alpha_0 +\alpha_{1i} x_1 +\alpha_2 x_{2i}+u_i$, and note that $Cov(x_{2i}, \widetilde{x_{1i}})=0$ because $\widetilde{x_{1i}}$ is the residual from an OLS regression with $x_2$ as a regressor. To proceed, you will need to make an assumption regarding $Cov(x_{1i}, e_i))$ and $Cov(u_{i}, e_i))$.
