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As far as I know, Superman has super powers because of Sun in our solar system. In Superman/Batman: Apocalypse he goes out of the solar system to fight Darkseid. I don't know if this happened before.
So, if he gets power from our Sun, how was he able to go outside of solar system?
EDIT:
I am not suggesting that Superman need a direct sun rays on his skin, but just to be "near" our Sun, i.e. in Solar System.
The short answer : Yes he does
To answer your question properly, we would have to know what Superman you are talking about. See As of 2012, how many different canon versions of the Superman character exist? for more details. If you are talking specifically about that Superman/Batman Apocalypse, then the question should be why does he keep his power, because he obviously does.
Anyway, I'm going to give you a few possibilities as to why Superman does keep his powers outside of our solar system. If you want more details, there are a lot of questions on this site about Superman’s powers.
First possibilities that come to my mind : Superman's powers come from a yellow star. Not particularly our sun, just a yellow one. And the range of a star is pretty good... The more bright is the star, the further its range would be. So even outside our solar system, either it is the Sun or another yellow star that gives him his power, Superman does keep them.
Second possibility : Superman's skin allow him to store energy. His skin stores the Sun's energy when it is exposed to its beams, and this energy can be used later. When there is no more energy, Superman loses his power. This is one of the answers to How did Doomsday killed Superman?
Superman gets his powers from a "yellow sun" not necessarily our sun (Sol). He absorbs the solar radiation to power his abilities, and stores it for later use. So as long as he's either near a "yellow sun" or doesn't expend his solar reserves he will be able to access his abilities. This is how he retains his abilities indoors, or at night time.
The only thing I would add to all the above, is that in some earlier stories, he either wasn't near a suitable sun or our sun became red and he used his Supermobile which provided him with all the power he needed.
in the comics it duplicated his powers and in the cartoon it provided him with power IIRC
Here's a link supermobile
Superman doesn't go out of super powers at night, it means that he doesn't need continuous solar radiation like Nuclear Man to have his super powers.
Superman can consume yellow radiation generated by anything which includes yellow stars other than Sun and even artificial objects.
In Justice League, once Toyman sent Superman in future when the Sun was red. In that situation, he got his super powers back from an artificial energy generator which was similar to Sun.
Such thing has been happened many times in the canon.
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.
In the limited comic series Superman/Aliens, Superman battles a colony of chest-burster Xenomorphs from the Alien film and comic series.
Most of the action takes place in deep space without the benefit of direct sunlight and Superman's power fades the longer he stays away from the sun. Eventually he is so weakened that aliens are able to impregnate him with a chest-burster of his own.
This was a cross-over series with Dark Horse Comics so it's probably not canon but it at least suggests that Superman stores power like a rechargeable battery and will slowly run down without solar energy.
In the Richard Donner film (1978) Jor'el said the yellow sun's radiation would affect his son's body chemistry giving him his powers once he was exposed- he didn't say he had to be exposed to the sun 24/7.
