The Prophet who built the Ark by Allah's command was Nuh (Noah, AS). The story of Nuh and the great flood is one of the most prominent narratives in the Qur'an, appearing in multiple surahs including Surah Hud (11:36-48), Surah Al-Mu'minun (23:27), and Surah Nuh (Chapter 71).
Nuh (AS) was sent to his people who had turned to idol worship. After centuries of preaching with little success, Allah commanded Nuh to build a large ark (ship). In Surah Hud (11:37), Allah instructed: "And construct the ship under Our observation and Our inspiration." The people mocked him for building a ship on dry land, but Nuh persisted in obedience to Allah's command.
When the flood came, Nuh boarded the ark with the believers and pairs of every animal species. The floodwaters destroyed the disbelievers, including Nuh's own son who refused to board. After the waters receded, the ark came to rest on Mount Judi (Qur'an 11:44).
The other prophets listed were not associated with building the Ark: Ibrahim (AS) is known for building the Ka'bah with his son Isma'il. Isa (Jesus, AS) performed miracles such as healing the sick and raising the dead. Musa (AS) is associated with the parting of the sea and the Exodus from Egypt.