What should a single woman do with 1 day in Nanjing while my partner is working.

I think you can explore the city for a day.
I suggest you to go to the Xuanwu Lake Park, Jiming Temple, Pesident Palace and Confucius Temple. At Confucius Temple area, you can also taste the local snacks.

Thank you.
Can I walk to all those places, or do I need a driver or taxi?

To reach Xuanwu Lake Park, I think local subway can help you. The subway line 1 can take you to reach Xuanwumen Station. Get out of the station, you can find the park.
Get out of the park from Jiefang Gate, you can walk for about 200 meters to find the Jiming Temple. Then you can take a taxi to reach Pesident Palace and then Confucius Temple.