Short answers, pulled from the story.
Octavian won the War of Actium by defeating Mark Antony's fleet on the 2nd of September 31 BC. His forces under Marcus Vipsanius Agrippa destroyed almost all of Antony's ships in open sea combat.
The decisive naval battle occurred on the 2nd of September 31 BC when Antony moved his fleet into open water. This engagement marked the turning point that led to the final defeat of Antony and Cleopatra.
Octavian blamed Cleopatra for the conflict rather than Antony to justify a declaration of war against Egypt alone. This strategy allowed him to frame the conflict as a defense of Rome against an eastern queen while avoiding direct legal attacks on a fellow Roman citizen.
Octavian countered with 350 Liburnian ships that were lighter and more maneuverable than Antony's 290 quinqueremes. These smaller vessels carried experienced sailors who had fought years of Roman naval combat unlike Antony's inexperienced crews.
Mark Antony fell on his sword on the 1st of August 30 BC according to Roman tradition after losing his army in Cyrenaica. He was left alive long enough from an open wound in his belly to be carried into Cleopatra's mausoleum before dying there.