Peter said that Jesus was raised up to David's throne by way of being raised up from the dead.
Acts 2:29 Men and brethren, let me freely speak unto you of the patriarch David, that he is both dead and buried, and his sepulchre is with us unto this day. 30 Therefore being a prophet, and knowing that God had sworn with an oath to him, that of the fruit of his loins, according to the flesh, he would raise up Christ to sit on his throne; 31 He seeing this before spake of the resurrection of Christ, that his soul was not left in hell, neither his flesh did see corruption. 32 This Jesus hath God raised up, whereof we all are witnesses. 33 Therefore being by the right hand of God exalted, and having received of the Father the promise of the Holy Ghost, he hath shed forth this, which ye now see and hear. 34 For David is not ascended into the heavens: but he saith himself, The Lord said unto my Lord, Sit thou on my right hand, 35 Until I make thy foes thy footstool. 36 Therefore let all the house of Israel know assuredly, that God hath made the same Jesus, whom ye have crucified, both Lord and Christ.
Notice here that Peter said there was a prophecy made by David where God told him that He would raise up Christ to sit on his (David's) throne and then said the prophecy was speaking "of the resurrection of Christ". And Peter said "This Jesus hath God raised up". By saying "This Jesus hath God raised up", he was saying that the prophecy about Jesus being raised up to sit on David's throne was fulfilled. God made Jesus "both Lord and Christ", proving that His level of power and authority had been elevated. Him being Lord means He is Master and King.
Like many OT prophecies, it has a spiritual fulfillment and not a literal fulfillment of Jesus sitting on a physical throne. You need to be willing to accept the NT explanations of the fulfillments of OT prophecies.