I concur with most posted answers above. Kalissar especially did a good job.
The only thing I would add is this:
1] If we are dealing with pre-crisis, not all of Superman's powers are solar related; some come from the difference in gravity between Earth and Krypton. Krypton is theorized to be between 33 to 150,000 times heavier in gravity than earth. As such, a kryptonian on any planet not similar in size and density could be anywhere from 33 to 150,000 times stronger, faster, and more durable than a human on earth, and any corresponding earth-like planet or environment.
2] We are uncertain of any other biological, chromosomal or evolutionary adaptations Kryptonians have as opposed to Earthlings... these could be something as mundane as night vision or a general broader visual spectrum, to a natural healing factor to antibodies which we don't possess which could affect their immunity and ageing. As such, a Kryptonian [Post-Crisis, and thus almost totally solar powered] without the yellow sun could, foreseeable, still be hundreds of thousands of years more evolved than humans that allows them to use subtler "enhanced human" abilities...think of Superman in a non-yellow sun environment being more like Captain America or a Wolverine than a human, but nothing near the God-like potential they have now.
3] It's been stated that different stars can affect his power levels differently. Most commonly, stars run from Red to Orange to Yellow to Green to Blue to White to Blue White. A chart for simplification follows here:
This basically relates to how much energy the star puts out. And Superman has been shown, in canon, to be empowered by those up to an Orange star, if only half as powerful as a yellow star makes him. Blue Stars have been shown to actually increase his powers, as shown here....
So it is not inconceivable that yes, he would have some form of abilities depending on where the environment is, but its most likely once there isn't a Red Star or or significantly heavier gravity present.
In the 2003 comic Birthright , they updated the origin story and gave a combination of the Gravity/Sunlight explanation, saying his cellular structure, which was essentially designed to absorb and draw some form of energy from an older, less powerful star, would essentially "drink in the radiation of a star so young." That, combined with his molecular density due to it being formed and evolved for a higher gravity environment means even without solar radiation, on a planet not as heavy or dense as Krypton, he'd be naturally stronger and more durable.