Can he reverse the flow of time for an environment outside of himself by traveling faster than light, (as he does in the Superman movie)? No. This was some bad hand waving done to rationalize even worse writing. Superman never had such a power in the comics.
Has he on the other hand traveled through time to different periods by flying faster than light? Yes. Pre-Crisis Superboy was a regular time traveler using the FTL method. But he never used this power to intentionally alter the past significantly if he could help it.
- In the past, (mostly in the Silver Age) when he traveled through time, the only person he was able to affect was himself. This was a particular iteration of Superman who was able to fly back and forth through time with relative ease. See: Time Trapper for an understanding of this particular version of Superboy.
One of the more noteworthy stories involving the Time Trapper came after the reality-altering mini-series Crisis on Infinite Earths and the 1986 revamp of Superman's origin, which removed Superboy from Superman's and the Legion's history. Given the problems this posed for Legion continuity, it was later revealed that the Time Trapper created a pocket universe from a slice of time in the distant past, and altered events in this reality so that an Earth resembling the pre-Crisis one was formed, complete with its own Superboy. The Time Trapper then further manipulated the timestream so that whenever the Legion would travel into the past to visit the 20th century (or Superboy visited the Legion's future), the two would be directed into each other's worlds. However, the "pocket universe" lacked a Kryptonian Supergirl, and thus was not a perfect answer to patching Legion continuity. (The "pocket universe" was later revisited and made the point of origin for a non-Kryptonian Supergirl, also known as Matrix).
- Depending on the writer, Superboy could take people with him through time if he carried them in a vehicle of some sort. In the picture below, he was taking his parents into the past to see pirates...
Travel through the fourth dimension, color coded for your convenience.
Most newer versions of Superman, as an adult, has had a limited capacity for time travel once writers got over the idea of time travel and had begun altering Superman's powers from their godlike state. Silver Age Superman, one of the mightiest versions of the character, was the last time Superman time traveled without a technological or magical device.
With that said, Superman #141 is one of the most famed of the FTL Superman's time travel stories where he travels back in time to visit Krypton before it's destroyed. He meets and interacts with his parents before the destruction of the planet.
Superman #141, November 1960, "Superman Returns to Krypton", written by Jerry Siegel.