First issue (pointed out in other answers) is gravity. I can see a single ring around a sun being feasible (an asteroid belt that became a planet?) but there are quite a few issues in a full sphere.
curious...How thick are you thinking the sphere would be?
The gravity has to be strong enough to hold the sphere together and spinning fast enough to give gravity, but not fast enough to send the spinning mass off into space.
Gravity will be strong in the equator where the spin is greatest, but almost non existent as you got to the top or bottom of the sphere
Our current understanding of how planets for unfortunately makes this setup very very unlikely if not impossible. There is no mass 'above' or 'below' the sun during solar system formation, it's kind of a '2d cloud' that planets form in around a sun. Admittedly this is drawing from an exceedingly narrow scope of knowledge that we currently have on planetary formations.
I really can't see how a system like this releases heat and energy. A sun (even a small one) puts out a lot of energy...in a standard case, a planet like earth radiates it back out to space. However, radiation in this sphere case would simply go back towards the sun. There would have to be a manner in which heat and energy travels through the shell of the sphere and releases out into space...and I have no clue how to do this in a manner that wouldn't melt the sphere.
Our sun is an incredibly strong magnet when it comes to it...only very recent research has been able to link the sun's magnetic cycles (sunspots in particular) to earth weather (yes, at the distance out sun is from earth, we see very consistent effects of the suns magnetic field from this distance). I'd be concerned that anything metallic that could be magnetized will be. This magnetism could be used to explain how the sphere holds together, but I would think any residents of the sphere wouldn't be compelled to use much for metal.
Reversely...the atmosphere has nothing to do with protecting earth from the majority of nasty space weather coming out of the sun. It's the Earths magnetic field that deflects this (around or through) the Earth. A planet needs a few elements to generate it's own magnetic field, nothing that a sphere could contain.
What happens if a Delta class sunspot throws a CME towards the sphere? At this distance, the atmosphere would actually fall apart at an atomic level and be forced to reform as the CME impact cascades through the atmosphere and into the land of the sphere itself.
I hate to say it...but with only an atmosphere to protect vs solar weather, there would be little chance of anything surviving (including water) on the inner surface of the sphere.
Unfortunately, had such a spherical world come into existence, I don't see it as 'stable' for long lengths of time. A single comet that would harmlessly plunge into the sun is now in a position to tear this sphere a completely new hole with all the matter from it either being flung into space or sucked into the center sun. Worse yet, after the hole is formed, there is nothing to hold in the atmosphere from leeching out as the entire system spins...and worse yet, this impact might very well collapse the whole system.