Answer by Ariel Williams, Dreamer, Writer, Artist
He would not just find it, but could go past it!
He actually flew past the end of the universe at least once! In one such case, Superman managed to break the barriers of the universe himself by flying faster and farther than he ever had before. (Pre-crisis) Supergirl was knocked unconscious while flying at her fastest. As she flew off into the unknown and possibly dangerous reaches, Superman flew off to save her. He followed her path exactly. He flew faster and faster smashing successive universal barriers until he smashed through infinity itself.<DC Presents #29 January 1, 1981 "Where no Superman has gone before">
He traveled so fast that he broke through the boundaries of the universe and smashed into the chest of a giant sized version of the. Not only did he not die, he could look back and see the entirety of the universe. In the above image, the whole of the universe is represented by a lemniscate (∞) or infinity symbol. The picture depicts Superman flying beyond the edge of the universe.
Superman: "In sanity's name what have I done?"
Superman: "I've got to turn around and return to my own universe before..."
Superman: "... but can I reach her before we're both too far gone to ever find our way back to our reality?"
In the DC universe, it is thought that the universe is infinite but there are many parallel universes in the greater multiverse. Superman has stepped across the boundaries into these universes numerous times. The boundaries or end of the universe can be found in black holes in ancient devices and in places where the space time continuum is in flux. So with that understanding not only could he find it, he has numerous times. He has even traversed the Marvel Universe.
As to die trying to find the end of his own universe physically? Well not necessarily, if he prepared properly.Superman Prime is the current Superman, but in the future, he will move his fortress of solitude into the center of the sun. He then meditates in his fortress for 15,000 years. When he came out, he was nearly infinitely powerful and immortal. He no longer was effected by red sun energy nor Kryptonite and did not need a yellow sun for power. This Superman could search for eternity for the end of the Universe and never die. He would likely outlive the universe itself, which in its own way might be a means of finding its end... More questions on Superman: