Learn Turkish: Bu sertifikayı aldığım sömestir çok yorgundum. - I was very tired the semester I got this certificate.

Turkish doesn't use a separate word for "when" here. Instead, it uses the object participle -dık (which softens to -dığ- here). Just as aldığım araba means "the car that I bought", aldığım sömestir means "the semester in which I received [it]". The participle itself links the action to the time word.

Why is sertifikayı in the accusative case?

Because it is the specific direct object of the verb inside the relative clause. You are talking about getting "this specific certificate" (bu sertifika), so it takes the buffer letter -y- and the accusative ending -ı.

Shouldn't it be sömestirde to mean "in the semester"?

You could say sömestirde, but it is not strictly necessary. In Turkish, when a noun phrase acts as a time expression for the main sentence (telling us exactly when you were tired), the locative suffix is often dropped. Leaving the time word bare like this is very common and natural.

Can you break down the endings on yorgundum?

It consists of the adjective yorgun (tired), the past tense copula -du (was), and the first-person singular suffix -m (I). Put together, it means "I was tired".