"I will explain the details at the attorney-general's office," he said.
"I want to surrender because I want the nation to develop and so that people can live peacefully."
Two military officials, speaking on condition of anonymity because the investigation is ongoing, described da Costa as a former policeman suspected in the shooting.
Ramos-Horta is recovering from the attack in a hospital in Australia.
Xanana Gusmao, the country's prime minister, who escaped unhurt in a separate attack the same morning, ordered the country's military and police forces to form a joint command to arrest followers of Alfredo Reinado, who led the fighters and was killed in the attacks.
Da Costa, also known as Susar, is among 17 people wanted in connection with the attacks.
He is the first to turn himself in to the joint command, surrendering in Turiscai, 120km south of the capital, Dili.
The military hopes that Gastao Salsinha, who took command of Reinado's fighters after he was killed, will also surrender.
Gusmao urged Da Costa and other fighters to co-operate.
"I am asking you to co-operate with the joint command so that people can live in tranquillity", he said at the government palace.
East Timor broke from Indonesia in 1999 following 24 years of occupation, declaring independence three years later after a brief period of UN administration.
The army split along regional lines in 2006, when about 600 soldiers were sacked, triggering factional violence that killed 37 people and drove 150,000 from their homes.