Apple - iPhone alarm through headphones only?

You could mute the voice before inserting the earphones and then put in the earphones and set the volume. This way the alarm sound will only come through the earphones because you have muted your phone, not the earphones.

No, unfortunately there is no such setting. Your best bet is probably to use a 3rd party app with alarm clock functionality. That way it will only play the sound through the headphones and not the speakers. The main disadvantage is that you would need to remember to keep the app launched before you go to bed each time.

I recorded a silent alarm and then used that as the sound and set the alarm to vibrate. This way, the vibration wakes me up but not everyone else. The vibration works really well if you put the phone next to or under your pillow. The phone can get warm if it is under the pillow though so I usually put it just to the side.