This is the lighthouse on the end of Oak Island that you can see from the pier at Southport and reach the lighthouse by way of the Oak Island Bridge. You can also see the Bald Head Island Lighthouse (Old Baldy) from here, but you'll need to catch a ferry to get to the island which is around $16 round trip.
The lighthouse is on Bald Head Island, and must be accessed by ferry from the mainland. It is lit at night, but not with a proper lighthouse type light, as it is no longer an active lighthouse. I understand it can be climbed, but I have not done that.
It is two nautical miles off North Carolina’s coast at the mouth of the Cape Fear River, Bald Head Island is a 20-minute passenger ferry ride from the mainland ferry terminal at Indigo Plantation and Marina.
From their webpage: For most of the year, ferries leave the mainland on the hour and the island on the half-hour beginning at 8:00 am. The winter schedule may be abbreviated. Advance ferry reservations are required, and should be confirmed three days in advance. (Call 910-457-5003.)