Archery and other forms of early artillery such as catapults were used in pre-cannon ships; however, the most common form of naval battle in the Mediterranean was ramming and boarding, with ranged weapons being used in a support role to soften up the enemy. Until the proliferation of deep water navigation, most navies consist of some form of galley: a low ship that had sails for regular travel, but large banks of oars for maneuver and propulsion during battle.

The galleys typically had a large bronze beak that would be used as a battering ram. The aim of the battle would be to maneuver one's ship so that you could ram the enemy's broadside, or sheer off the oars poking out of the side, which would cause much pain and suffering to the rowers on the other end, and also slow down the ship making it meat for a second pass.

The other method is boarding. The galley would hold a number of heavy infantry that we today would call marines. The galley would attempt to maneuver into range of an enemy, and then latch on to it with planks, ropes or specialized devices like the Roman corvus, which was a sort of boarding bridge that could crash onto the enemy deck and latched on with hooks. The heavy infantry would then rush on and turn it into a melee. Typically, this was the tactic preferred by the more land-minded party who was trying to usurp a traditionally naval force; i.e. the Romans against the Carthaginians, the Spartans against the Athenians, and the Holy League against the Ottomans at Lepanto.

So yes, archers were part of this process, but a lot of times naval battles were decided by tiny land engagements on small shaky platforms where a misstep meant certain, horrible death by drowning.