This is profoundly silly, but I love your illustration so I am going to have a go at this.
As drawn, the ball with all the stars on is small compared to the void in the sphere. Either the sky would be filled with the view of the other side of the earth, or light does not travel in straight lines. Let's say light does not travel in straight lines.
Let us take our universe, centre it on the centre of the Earth, and transform it so R maps onto 1/R with R=1 being the average surface of the Earth. The centre of the earth is now infinitely distant, and the distant stars are now a very small cloud of objects close to the centre of the sphere/universe. If you look up, your light of sight bends to point towards this centre.
The Moon and the Sun are still spheres with this projection: a sphere maps onto a sphere. They are not quite the same spheres as the features on the near side will be slightly larger than the features on the far side, but this distortion would be small for anything that has little visual depth for the observer.
The Moon would go in a circle. The Sun would go in a smaller circle. The planets will orbit the sun in circles that aren't centred on the sun. The physics behind this will be hard to formulate in simple laws, because we have a lot of visual depth. Take Jupiter, for example. That is somewhere between 4.2 to 6.2 AU, where the Sun is close to 1.0, so Jupiter will move in a circle that is about a fifth the distance from the centre of the universe as the sun. We do not see these circles from the surface of Hollow Earth - we see the sun and the planets as we see them on Sensible Earth.
The one thing I cannot manage is a Lunar eclipse. We can't see the shadow of the Earth on the moon.