Get aboard the ship
Exit the cave after you see the cutscene. After the phone talk fast travel to Badtown. Once there, go to the bar, where you will meet Buck. Leave the bar and make your way to the eastern shore - it's best to use the pier to do that. There you will find a jet ski that you need to take to approach the ship (don't get too close) that is your current objective.
When you come close enough, make sure that there are no sharks in the water and cover the rest of the distance by swimming /diving, so that you are not noticed by your enemies onboard. To go aboard, you need to take the ladder on the ship's left side (in relation to the ship's stern).
There are quite a few enemies onboard. You can try and avoid encounters with them, or eliminate them quietly. If you have decided on killing them out, remember that there is an armored enemy out there and, as yet, you cannot kill him using Takedown.
Find the Laptop Computer On Nostalgia's Board
After you enter, right behind the door, there will be the first one of the pirates waiting. Kill him without raising the alarm.
Further on, there will be another two waiting. Lure one of them away, by tossing a stone, so that you can kill him out of the other guard's sight.
Kill the other guard whenever it is convenient and neutralize the alarm on the wall. Climb down the stairs and go through the door.
Right to your right, there will be another enemy. Wait for him to look away from the door that you entered through and kill him quietly. Go on and kill another enemy in the very same room. Go through next door, and lure the pirate away by tossing a stone, out of his comrades' sight. Dispose of the rest of the enemies in the same way.
Open the massive door, behind which there is the armored enemy standing, with his back at the door. Shoot him on the nape of his neck to get rid of him quickly. Use the laptop computer on the desk.
Get Out of the Sinking Ship
Swim towards the designated area, and use glowing oxygen tanks on your way, to replenish your oxygen supply. Swim through the passageway and take a turn to the left, and then, to the right at the end of the corridor. You will be now a able to get up on the part of the deck that's above the surface of the water. As you pass through consecutive rooms, you will, eventually reach, a door which you need to open.
The room will be flooded with water, as a result. Swim through the opened door and try to get lower through the passageway to the left. You will be attacked by a pirate. Press the sequence of jump, crouch, use, to get rid of him.
Swim into the corridor behind the pirate's corpse, where you will find another oxygen tank that you need to use. Further on, swim over to the right side and take the corridors on.
You will, eventually reach a room with a tank, and a window that needs to be smashed. Once you do it, you will be able to get outside, which will complete this mission.