Learn Polish: Ta nowa instytucja jest administrowana przez stary teatr. - This new institution is administered by the old theatre.

This is a key feature of the passive voice! Even though in the active voice you would say administrować tą nową instytucją (Instrumental), when you make the sentence passive, the object receiving the action becomes the subject. As the subject of the sentence, it must take the Nominative case, becoming ta nowa instytucja.

How exactly do we form the passive participle administrowana from administrować?

For verbs ending in -ować, you drop the final -ć and add the passive endings -ny, -na, or -ne. Because instytucja is a feminine noun, we need the feminine ending -na, which gives us the form administrowana.

Since the verb is related to the Instrumental case, why do we use przez to say "by the old theatre"?

In Polish, the agent of a passive sentence—the one actually doing the action—is almost always introduced by the preposition przez followed by the Accusative case. This rule applies no matter what case the verb governs in its active form. Here, stary teatr is in the Accusative (which happens to look identical to the Nominative because it's an inanimate masculine noun).

Is administrować the most common verb for "to manage" or "to administer" in Polish?