Here is what I have so far:
\begin{align} \Lambda &= \frac{L(\lambda_1)}{L(\lambda_0)} \\[8pt] &= \frac{\prod \frac{\lambda_1^{x_i}}{x_i!}e^{-\lambda_1}}{\prod \frac{\lambda_0^{x_i}}{x_i!}e^{-\lambda_0}} \\[8pt] &= \left(\frac{\lambda_1}{\lambda_0}\right)^{\sum x_i}e^{-n(\lambda_1-\lambda_0)} \end{align}
We see that the Poisson family has a MLR in $T(X) = \sum x_i$. By Karlin-Rubin theorem, the test with rejection region $C=\{T(X) > t_0\}$ is UMP level $\alpha$ where $t_0$ satisfies $P_{\lambda_0}(T(X) > t_0) = \alpha$. But now I'm stuck.
How do I find an expression for $t_0$ using $P_{\lambda_0}(T(X) > t_0) = \alpha$?