Sunday night solo dining in Charlotte
For solo dining, I might first recommend sitting at the bar at 5Church. The bar at Rooster's would be good as well.
Other places downtown where you could feel comfortable sitting at the bar by yourself are Enso for sushi and Asian cuisine, Vida for "upscale" Southwestern/Mexican, and Dandelion Market for decent small plates.
You could also sit at the bar at Halcyon, where you could get great food, but I don't know if it's the best place to dine solo on a Sunday.