One can play around with the rotation of the spherical projection when creating the UVs but you will get distortion at the poles. The problem is the texture that you are using. The textures is for a spherical projection where the north and south poles are at the top and bottom. The texture compensates for the projection distortion.

You will need to create your own texture where Mecca is at the top.
I created the image below by doing the following:

1. Create a sphere and place it at the origin.
2. Flip the normals on the sphere.
3. Assign a lambert with color set to white to the sphere.
3. Create a point light at the origin.
4. Mapped the existing texture to the color channel of the point light.
5. Rotate the point light until the projected image is where you want it. I did this by trial and error with my render camera inside the sphere facing one of the poles. I'm sure there is some mathematical calculation that one can use. Maybe our resident surveyor can help
6. Bake out the color and light to a texture. Use a 2:1 ratio e.g. 1024x512.

You can use the created texture in a standard spherical projection.

Earthmap courtesy of visibleearth.nasa.gov.