This is going to be harder because of the time period. The information I have should work for people who arrived from about 1880-1930, but earlier records are harder to find or simply don't exist. But here are the steps anyway, you can give it a try.
Your best bet is to start with the census. 1840 or 1850 or 1860, if they are accessible. You need to find her parents' names. If you know her surname and the names of her siblings and a city, you should be able to match up the family. Once you know her parents' info, you can try to find their naturalization records. This will have the ship name and date, then you can look up the manifest.
My husband's grandfather was also born at sea in 1899. From what I can tell, this made him a US citizen by birth and he did not have to naturalize. Your gggma would undoubtedly also be American by birth, since laws got stricter with time, not more permissive.
Also try to find your gggma's marriage certificate and death certificate. These usually have parental names and sometimes the mother's maiden name or places of birth. Obituaries can be very helpful as well. Look for your gggma and for her siblings, if you know them.
This is doable but it will require some careful research. Even more importantly, careful record keeping, Every single little fact is potentially important, even if it seems too trivial to record. Also record where you got the information. Your sources are very important and will help you sort things out later (you will have a lot of facts that contradict each other and that's okay, just note them all).