If, sometime during your last trimester visit, you find out that your baby is in the occiput posterior position, facing your abdomen instead of your back, does that mean you are "doomed" to go through back labor?

No.

You might be lucky. I am currently on my 38th week of gestation, and during my recent check up, I was told that the baby has shifted positions. I'm hoping that it will remain in the face-down position until delivery time.

If your baby doesn't appear to change positions, take heart. The baby can still rotate to face downwards at the last minute, just as the baby passes through the birth canal. And even if it doesn't, don't despair. There are ways to manage pain to make back labor more bearable. I personally found going on my hands and knees plus a massage or counterpressure on my spine at each contraction quite helpful in relieving pain. A warm soak in the tub also did great wonders!