Learn Arabic (Modern Standard): نحن في مركز المدينة.

It comes directly after the preposition في (in). In Arabic, nouns that follow a preposition must take the genitive case (المجرور). For most singular words like مركز, this is marked by adding a kasra (an "i" sound) to the final letter, making it "markazi".

We translate this as "the center of the city", so why doesn't مركز (center) have ال (al-) attached to it?

The phrase مركز المدينة (the center of the city) is an Idafa construction, which shows possession or a close relationship. The golden rule of Idafa is that the first word (the thing being possessed or specified) can never take the definite article ال, even though it translates with "the" in English.

Does المدينة (the city) also take a kasra at the end?

Yes, it is pronounced with a kasra at the end: المدينةِ (al-madinati). The second word in an Idafa construction is always in the genitive case. So in this sentence, مركز is genitive because of the preposition, and المدينة is genitive because it is the second part of the Idafa.

Where is the word for "are" in this sentence?

In Arabic, simple sentences in the present tense do not use a "to be" verb like "is" or "are". You just place the subject نحن (we) and the information about it في مركز المدينة (in the center of the city) next to each other, and the "are" is naturally understood.