T'hillim / Psalm 24:1 begins with the words "Of David, a song." Again, most Christian translations either eliminate the first line (KJV for example) or they don't number it.
The NIV and NRSV Christian translations retains those first words, but shows them as a "header" -- they are not shown in the body of the psalm which could mislead readers into not seeing them as PART of the psalm. The King James Version eliminates those words altogether!
To claim this psalm is about Jesus when the very first line tells us that is a song of and about David is deceitful at best. The King James eliminates the word "of David a song" altogether!
Let's read the first four lines in context: "Of David a song. The land and the fullness thereof are the L-rd's; the world and those who dwell therein. For He founded it upon seas and established it upon rivers. Who will ascend upon the L-rd's mount and who will stand in His Holy place? He who has clean hands and a pure heart, who has not taken My name in vain and has not sworn deceitfully." T'hillim / Psalm 24:1-4
Having now read the first four lines of the psalm IN CONTEXT please answer the question: can this be a prophecy predicting the exaltation of Jesus?
It is a psalm about King David (line 1) who is praising G-d (lines 1-2) and stating that good people can be close to G-d (lines 3-4).
It says nothing about the messiah.
It does not predict the exaltation of the messiah.
The missionary lifts the sentence "Who will ascend upon the L-rd's mount and who will stand in His Holy place?" out of context and hopes the read ignores the very next words: "He who has clean hands and a pure heart, who has not taken My name in vain and has not sworn deceitfully." T'hillim / Psalm 24:1-4
Those who are still unsure -- perhaps G-d is speaking only of Jesus -- read the next few lines and it is clear that David is speaking of righteous people -- not about one specific righteous person who is exalted as is claimed by the list maker. "This is the generation of those who seek Him, who seek Your presence-Jacob, forever. [You] gates, lift your heads and be uplifted, [you] everlasting portals, so that the King of Glory may enter." T'hillim / Psalm 24:6-7
In the psalms, primarily written by King David and many autobiographical, David often speaks about righteous people who through their righteousness are close to G-d. T'hillim / Psalm 33:1 says "Sing praises to the Lord, O you righteous; for the upright, praise is fitting." T'hillim / Psalm 65:5 says "Praiseworthy is he whom You choose and draw near to dwell in Your courts."
The supposed "proof texts" which the list maker claims proves that Jesus the psalm predicted that Jesus exaltation was predicted says do nothing of the sort. Acts 1:11 says "Men of Galilee,” they said, “why do you stand here looking into the sky? This same Jesus, who has been taken from you into heaven, will come back in the same way you have seen him go into heaven.”
This passage claims that Jesus was taken to heaven -- but it does not do anything to prove that he was -- or that his "exaltation" was predicted in T'hillim / Psalm 24:4.
The list maker also presents Philippians 2:9 as proof that the psalm predicted Jesus' exaltation. Philippians 2:9 says "Therefore G-d exalted him to the highest place and gave him the name that is above every name." Again -- this passage is in the Christian bible but where is the proof? The author makes this claim but one could as easily claim that Mohammad or Buddha was exalted to the highest place. Would a Christian be expected to believe the claim about Mohammad or Buddha? If not, why not -- if they believe such a claim made and not proven by the anonymous author of Philippians (supposedly Paul who never even met Jesus)?
Psalms are poems, prayers -- they are not prophecy. This psalm is recited often in Jewish religious services. The first words of this psalm are reversed from T'hillim / Psalm 23 -- לְדָוִד מִזְמוֹר (of David a song) in this psalm and מִזְמוֹר לְדָוִד (a song of David) in T'hillim / Psalm 23. Why? The sages discussed why the words were reversed. . . The Rabbis said: Wherever it says: “A song of David,” he would play his musical instrument [Harp] and he would find himself inspired by G-d. The song connected him to G-d. Wherever it says: “Of David, a song,” David was inspired by G-d first and then he recited a song.
Rabbi Mordechai Torczyner has an excellent audio lesson on this psalm available at Yeshiva University Online (link).