Short answers, pulled from the story.
Vladimir Putin became Acting President immediately on the 31st of December 1999 following the resignation of Boris Yeltsin. This transition occurred before the presidential election held in March 2000.
The first inauguration ceremony for Vladimir Putin took place at the Grand Kremlin Palace rather than the originally planned State Kremlin Palace. The event lasted exactly one hour within these historic halls and marked a change since the Soviet collapse.
Marat Baglai served as the President of the Constitutional Court of Russia and rose on the podium to oversee the transfer of power. He called on Vladimir Putin to take the oath alongside other officials including Yegor Stroyev and Gennady Seleznyov.
Vladimir Putin left St. Andrew's Hall accompanied by the song Glory composed by Mikhail Glinka followed by an artillery salute. An artillery salute also occurred while Boris Yeltsin introduced the Kremlin Regiment to him at the Cathedral Square of the Moscow Kremlin.