If the Hebrew Bible gets read in churches on Palm Sunday, it will probably be Psalm 118, which celebrates some Hebrew victory. Since the Psalm seems to come from the 2nd Temple period, the only victory would be that of the Maccabean revolt.
The reason this Psalm gets read in connection with Jesus entering Jerusalem is that it describes a victory celebration entering Jerusalem (vs. 18-20).
Also v. 27 can be translated, as the New Revised Standard Version has it, “The LORD is God, and he has given us light. Bind the festal procession with branches, up to the horns of the altar.” The mention of branches fits Palm Sunday when people strew branches before Jesus.
Palm branches carried by children often give a visual expression of this in our churches. (A historical note: only John 12:13 says that the branches were palm branches. Palms trees don’t grow at the altitude of Jerusalem, so, if the crowd had palm branches, they would have had to carry them from Jericho or somewhere else in the Jordan valley.)
This Psalm is well loved because it praises the “steadfast love” of God, thus contrasting God’s love with human love that we “fall” into and out of.
The Psalm also contains the popular verse, “This is the day which the LORD has made; we will rejoice and be glad in it” (vs. 24).
Furthermore, it has “The stone which the builders rejected has become the head stone of the corner” (vs. 22). This is the basis for the Christian use of the Psalm to refer to Jesus.
In the 1906 A critical and exegetical commentary on the book of Psalms, which you can find online here, Charles and Emilie Gace Briggs explained,
“Zion is the cornerstone of the Kingdom of God in accordance with Is. 28:16. The nations had done their best to destroy it. The last effort had been made by Antiochus, the king of Syria, but in vain. He had been overcome. Zion had regained her strength and glory through the victorious armies of the Maccabeans, and the omnipotent right hand of Yahweh. The Messianic application of the passage is due to the fact that the person of the Messiah bears the same relation to a kingdom of living persons that Zion, the capital of the kingdom, does to the kingdom” (p. 407).
In other words, the kingdom is more a community centered around Jesus for Christians than a polity with a seat of government in a particular place.
Anyway, I would say that for both Jews and Christians, Psalm 118 celebrates the faithfulness of God. I will say again that the Psalms are songs and meant for singing. You just don’t get them if you read them as prose, and especially if you read them as a code of dry religious tenets.