This reminds me of bicycle chains and gears. How long does a chain have to be to reach around two gears? How many teeth fit into the chain as it wraps around?
But that sunrise will not be a little Sun coming up in the East. It will look more like a total solar eclipse fading away. A bright ring will surround the edge of the comet and slowly fill your sky to the horizon as you get closer. You would be able to see the corona first, followed by the chromosphere, and finally the bright photosphere. This effect, also called the two-pulley problem, means that comets absorb sunlight over a larger area close to the Sun than they do far away. This helps them sublimate more rapidly.
Life on a sungrazing comet would sure be different!
Check out our 3-D rendering of the orbits of sungrazing Comets ISON and Lovejoy. It shows both orbits near perihelion. You can explore why different satellites see comets differently.