To answer my own question now that my trip is over...
There was an additional restriction of "not having traveled outside of China in the last
40 days 14 days" (I thought I heard “40 days” but it was probably “14 days” in hindsight; they sound similar in Chinese: 十四 vs. 四十). I boarded the plane at Beijing Daxing airport and they took me aside and checked my passport for stamps. I did not see any signage anywhere throughout the trip.
As usual nowadays in China, there was also a QR code that required scanning before entering airports (which turns green if you're okay to go). Since Beijing and Ningbo use different systems, I was a bit confused.
Aside from that, domestic plane travel within China was as usual (except to wear a face mask when possible). It didn't take substantially longer.
Some other points:
Staff at Beijing Daxing airport were far more rigorous, and there were more questions than at Ningbo airport. In fact, at Ningbo airport, about 30% of people weren't wearing face masks around the airport (which is basically unthinkable in Beijing).
My impression is that the airport staff cannot easily explain the (new) regulations in English. I can speak some Chinese, which helped.
There were some additional measures like temperature checks, and hand sanitizer was provided. (All very routine nowadays.)
The hotel may have had its own restrictions in place. To be safe, I went for a coronavirus test shortly prior to my trip (it was 70 yuan at one of several possible hospitals in Beijing, and it was done out the front of the hospital---the test results were reported on WeChat, so I only needed a single trip; it took me a long time to figure out what to actually do). When I checked into the hotel in Ningbo, I was feeling tired so I just gave them my phone to check; I'm not sure what they actually checked. We didn't need to wear a face mask around the hotel in Ningbo (in fact, wearing face masks was rare throughout Ningbo).
After I got back, the WeChat app on my phone (健康宝 used in Beijing) stopped working until I logged in where I had traveled. (I didn't realize this until I tried to enter a building.)