Watch

To get a watch you must first have a coordinate based clue scroll, you can then talk to the Observatory Professor. Ask the professor to help you and he will tell you what to do.

Watch: Speak to Brother Kojo in the in Clock Tower south west of Ardougne's zoo.

Sextant: Speak to Murphy in Port Khazard, located north-east of Yanille.

Navigation chart: Talk to the Observatory Professor and he'll tell you how to use the sextant then will give you a Navigation Chart.