EDIT: Whoops
How can the sample mean be 2.42? That implies $\displaystyle \sum X_i = 24.2$ which is impossible.
Anyways, $\displaystyle P_{\lambda = 1.5} (\sum X_i > 24.2) = .01$
so yes, you would. I used R to calculate the probability, since it's a tedious calculation to do exactly (essentially summing the pmf). In other words, the appropriate k that gives the size of desired test is less than 24.2.
I don't worry about you having to calculate UMP tests in real life because, in general, they don't exist Even throwing in a single nuisance parameter is typically going to ruin things for you, so most of the tests that people use in practice aren't UMP.