Historically Zerubbabel did not complete God's house, namely the
restored Temple of Jerusalem. It was finished in a later generation
during the time of Ezra and Nehemiah. Ezra 4 describes how Zerubbabel
refused help from fellow would-be-worshipers (apparently the future
Samaritans), saying: . . .
There is no evidence to support this allegation. In fact, Zerubbabel was a contemporary of Ezra and Nehemiah: he did not live in an earlier generation prior to them.
The temple rebuilding was completed, start to finish, within a 49-year period of time, well within the lifetime of a single man and not requiring multiple generations. The prophecy of Daniel 9 marks out the seven-week, representing 49 years, period of time for the restoration of the temple. And the Jews declared how many years of this were occupied in the actual construction.
Then said the Jews, Forty and six years was this temple in building,
and wilt thou rear it up in three days? (John 2:20, KJV)
(Several years more involved delays over the legal hassles imposed by the enemies of Jerusalem.)
One possible clue to the progress and work of the temple is that Zerubbabel also went by another name, in similar fashion to the Babylonian names given to Daniel and his three friends. Zerubbabel's other name was "Sheshbazzar." These names are used somewhat interchangeably, as can be seen in the following verses.
Then rose up Zerubbabel the son of Shealtiel, and Jeshua the son
of Jozadak, and began to build the house of God which is at
Jerusalem: and with them were the prophets of God helping them.
(Ezra 5:2, KJV)
Then came the same Sheshbazzar, and laid the foundation of the
house of God which is in Jerusalem: and since that time even until
now hath it been in building, and yet it is not finished. (Ezra 5:16,
KJV)
(And notice, too, that these parallel verses are both in the same chapter of Ezra. But this connection is confirmed in the prophecy under study.)
The hands of Zerubbabel have laid the foundation of this house; his hands shall also finish it; and thou shalt know that the LORD of
hosts hath sent me unto you. (Zechariah 4:9, KJV)
Notice, too, the position each is designated.
Even those did Cyrus king of Persia bring forth by the hand of
Mithredath the treasurer, and numbered them unto Sheshbazzar, the
prince of Judah. (Ezra 1:8, KJV)
And the vessels also of gold and silver of the house of God, which
Nebuchadnezzar took out of the temple that was in Jerusalem, and
brought them into the temple of Babylon, those did Cyrus the king take
out of the temple of Babylon, and they were delivered unto one, whose
name was Sheshbazzar, whom he had made governor; (Ezra 5:14, KJV)
And the LORD stirred up the spirit of Zerubbabel the son of
Shealtiel, governor of Judah, and the spirit of Joshua the son of
Josedech, the high priest, and the spirit of all the remnant of the
people; and they came and did work in the house of the LORD of
hosts, their God, (Haggai 1:14, KJV)
According to the Bible, the temple was completed during the time of Haggai and Zechariah and Nehemiah.
14 And the elders of the Jews builded, and they prospered through the
prophesying of Haggai the prophet and Zechariah the son of Iddo. And
they builded, and finished it, according to the commandment of the God
of Israel, and according to the commandment of Cyrus, and Darius, and
Artaxerxes king of Persia. 15 And this house was finished on the third day of the month Adar, which was in the sixth year of the reign of Darius the king.16 And the children of Israel, the priests, and the Levites, and the rest of the children of the captivity, kept the dedication of this house of God with joy. (Ezra 6:14-16, KJV)
So what began in the time of Cyrus was completed in the time of Darius. And Zerubbabel was clearly the one who was building during the time of Darius.
In the second year of Darius the king, in the sixth month, in the
first day of the month, came the word of the LORD by Haggai the
prophet unto Zerubbabel the son of Shealtiel, governor of Judah, and
to Joshua the son of Josedech, the high priest, saying, (Haggai 1:1,
KJV)
Conclusion
It was indeed Zerubbabel in charge of the work when the rebuilding of the temple was completed. Zerubbabel, also known as Sheshbazzar, was a contemporary of Haggai, Zechariah, Ezra, Nehemiah, Cyrus, and Darius, as the Biblical record indicates. Therefore, the prophecy of Zechariah 4:9 was fulfilled during his time, as promised.