Liz,
Statutory birth registration only started in Ireland in 1864 so you won’t find a birth certificate for Robert. You might find his baptism though but to do that we need to know his denomination.
Looking at the 1901 census of Co Down, there were 317 people named Andrews in the county at that time. The majority were Presbyterian (reflecting the fact that most of the population originated in Scotland) but other denominations are found too.
If Robert was Presbyterian, there are 6 Presbyterian churches in Bangor. 1st Bangor has baptisms from 1852, 2nd 1829, Ballygilbert 1841, Ballygrainey 1830, Conlig 1845 & Groomsport from 1841. There are copies of these records in PRONI (the public record office) in Belfast, but they are not all on-line so far as I am aware and you would probably need to get a researcher to go to PRONI to check them all for you.
Andrews is a pretty common surname in Co. Down and ideally you ought to try and find his parents names from a marriage or death certificate in order to be sure you have found the right man. His date of birth may not be 100% accurate. (People in Ireland in the 1800s didn’t celebrate birthdays and didn’t always know their date of birth and so sometimes just made a date up when officialdom asked for one). Also I notice he was just 15 when he joined the Navy. If the Navy had a minimum recruitment age people sometimes “adjusted” their age to satisfy that requirement.