I suspect you usually think of "obey" as "following a code" because you grew up in a culture where you obey a systematized set of rules imposed by an impersonal government. In first-century Judea, obedience was always to a personal someone, whether your parent or governor or a spirit (who then obeyed someone above them). Obedience in the Acts 12 passage is little different than a Spanish-speaker today saying "mande" ("order me") in response to an authority getting their attention. "Faith" is only part of that category system in the sense of "faithfulness".
"ὑπακούω" has "ακούω" (to hear) as its primary semantic root. Do a word study of that in the Biblical texts and you will quickly find that "hearing a person" implies "obedience to that person" to a far greater degree than in Western societies today, especially in the OT. "Hearken" is about as close as we can get in English today.
Indeed, you might read Julian Jaynes' seminal work "The Origin of Consciousness in the Breakdown of the Bicameral Mind" to start getting some sense of the strength of that connotation: he posited that preliterate societies often interpreted messages going from their right brain to their left as the voice of the king, or god, which could not be disobeyed any more than modern schizophrenics can disobey such messages.