If it doesn't work out at the airport, go to a Unicom customer service center in the city, not one of the small shops you see everywhere with a Unicom sign out front. Though they can sell you a SIM card, the small shops can't activate the card if you're registering with a passport. That's why they send you to another place for name registration and activation.
Even small unicom stores can sell and register new SIMs, but they generally can't put your old number on a new SIM. I avoid the service centers at all costs.