Aurelion Sol's damage mechanics rely almost entirely on his ability to position himself well from moment to moment. The Movement Speed burst from Phase Rush lets you chase and stay in perfect range for follow-up [W] hits.
Sol's massive lac of mobility makes him incredibly vulnerable to jungle ganks and mage assassins like LeBlanc. Nullifying Orb helps you survive those ganks and all-ins so that you can scale into the mid game without being way behind.
Again, mobility is the name of the game for Sol. Combined with his [E] passive Movement Speed, Celerity will grant you very reliable bonus AP.
Waterwalking is a great rune to take on Sol when you're looking to get the early shove and roam to the side lanes for map pressure.
Minion Dematerializer is fantastic for empowering your early waveclear so that you're able to begin snowballing side lanes.
Cosmic Insight's general utility is fantastic for teamfight where you are looking to get multiple stuns from your [Q].