Learn Russian: Первое здание — школа, а девятое — наша больница. - The first building is a school, and the ninth is our hospital.

Ordinal numbers act exactly like adjectives in Russian. They must match the gender, number, and case of the noun they describe. Because they describe здание (building), which is a neuter noun in the Nominative case, we use the neuter forms первое and девятое instead of the masculine dictionary forms первый and девятый.

Why is there no noun after девятое?

Just like in English, you can drop the noun if it is already obvious from the context. Here, девятое means "the ninth one" and stands in for девятое здание (the ninth building). Since the implied noun is still the neuter здание, the ordinal number stays in its neuter form.

If школа is feminine, why does the sentence start with the neuter word первое?

The ordinal number первое modifies the subject здание (building), which is neuter. The fact that this building happens to be a школа (a feminine word) does not change the grammatical gender of the word здание. Modifiers always match the specific noun they are attached to!

Why use the conjunction а here instead of и for "and"?