The governor of Jerusalem who helped rebuild the wall of the city (Neh 1:1; 8:9; 10:1; 12:26,47). Nehemiah was a descendant of the Jewish population that had been taken captive to Babylon in 587-586 B.C. In 539 B.C. Cyrus the Persian gained control over all of Mesopotamia. He permitted the Jewish exiles to return to the city of Jerusalem. Nearly a century later, in Nehemiah's time, the Persian ruler was Artaxerxes I Longimanus (ruled 465-424 B.C.). Nehemiah was his personal cupbearer (Neh 1:11).
In 445 B.C. Nehemiah learned of the deplorable condition of the returned exiles in Jerusalem (Neh 1:2-3). The wall of the city was broken down, the gates were burned, and the people were in distress. Upon hearing this, Nehemiah mourned for many days, fasting and praying to God. His prayer is one of the most moving in the Old Testament (Neh 1:5-11).
Nehemiah then received permission from Artaxerxes to go to Judah to restore the fortunes of his people. He was appointed governor of the province with authority to rebuild the city walls.
Once in Jerusalem, Nehemiah surveyed the walls at night (Neh 2:12-15). He gave his assessment of the city's condition to the leaders and officials and then organized a labor force to begin the work.
Nehemiah and his work crew were harassed by three enemies: Sanballat the Horonite (a Samaritan), Tobiah the Ammonite official, and Geshem the Arab (Neh 2:10,19; 6:1-14). But neither their ridicule (Neh 4:3) nor their conspiracy to harm Nehemiah (Neh 6:2) could stop the project. The builders worked with construction tools in one hand and weapons in the other (Neh 4:17). To the taunts of his enemies, Nehemiah repled: "I am doing a great work, so that I cannot come down" (Neh 6:3). Jerusalem's wall was finished in 52 days (Neh 6:14)-a marvelous accomplishment for such a great task. Nehemiah's success stems from the fact that he kept praying, "O God, strengthen my hands" (Neh 6:9).
Nehemiah's activities did not stop with the completion of the wall. He also led many social and political reforms among the people, including a return to pure worship and a renewed emphasis on true religion.
(from Nelson's Illustrated Bible Dictionary, Copyright (c)1986, Thomas Nelson Publishers)
...why not just read the book of Nehemiah for yourself?
Regarding the Holy Spirit...
You've asked specifically for restoration work, so i'll leave the Gifts and all out of it.
Basically the HS does a two-fold work: Life and Power.
Life, that 'restoration work' you're asking about is salvation itself. The HS is breathed into a person (John 20:22) and that person then has life in the Spirit (Romans 8), especially (Romans 8:9-17), which grows (Matthew 13:31) and bears fruit (Matthew 13:23), (Gal 5:22).
This is distinct from the power the Spirit brings, to be witnesses and to do the good works on earth. (Acts 1:8), and manifests in Gifts, etc.
Or were you referring to the ministry the HS does for Christians (e.g. Comforter, Teacher, etc?)