I'll give it a try, but since I myself am a student, I cannot guarantee my reply is the correct one. It may bump up the thread however to the attention of experts.
If I am testing against , I will use the likelihood ratio that you already stated:
This shows to me how likely the Ho compared to H1. To be considered a test, this ratio should be compared to some number (k) that is determined by the confidence level of the test (alpha):
Then I'll just substitute theta's and n that I am given and find the expression for the test statistic vs k:
After some manipulations (in particular, taking log, which allows to keep the same sign of the inequality, and simplifying algebraically)
that will be your
The last question 'optimal property that this test possesses' I am not sure what exactly this is supposed to mean. I think it should be clear from the focus of your study guide, or other similar questions in the textbook.