Camotes can be reached from Cebu port sailing with OceanJet. The ticket price is 500 pesos for an open-air/tourist class and 800 pesos for business class.
The ferries depart daily from Pier 1.
Or you travel to Danao port.
That is a little bit more complicated. You can use a taxi but they will charge you maximum for it is outside the city and they go back empty.
If you can convince a rider to use the meter, it also will be expensive due to heavy traffic on the highway.
Better travel from the North bus terminal, taking any bus passing by Danao (direction of Maya)
Tell the ticket man that you need to get to Danao Port, they will drop you off in the right place.
It should be before reaching the Danao public market, once you are there, you have gone too far.
From the bus stop at the port area, it is only a small ride, you can stop a tricycle and ask them to take you to the passenger terminal.
Here are the 2018 ferry times for both Cebu port and Danao port:
Super Shuttle Ferry 05.00 Danao Port, daily
Jomalia Shipping 05.30 Danao Port, daily
OceanJet 06.00 Cebu City Pier 1, daily
Jomalia Shipping 08.30 Danao Port, daily
Golden Shipping Fastcraft 09.00 Cebu City pier 1, daily
Jomalia Shipping 11.00 Danao Port, daily
Jomalia Shipping 14.00 Danao Port, daily
OceanJet 15.00 Cebu City Pier 1, daily
Golden Shipping Fastcraft 17.30 Cebu City Pier 1, daily