The answer to this question depends in part on how we understand OT prophecy.
I don't read Isaiah 53 as a future prediction of Jesus, in the sense that Isaiah had a mental picture of Jesus hundreds of years before Jesus was even born. Rather, Isaiah gave us a picture of a suffering servant through whom God would bring about the healing of the whole world. Isaiah was inspired to that picture in part by his meditation on the Jewish sacrificial system, and so he uses that language to paint his prophecy. The imagery of sprinkling comes from the same place. Consider for example Leviticus 16.18-19:
“Then he [the high priest] shall come out to the altar that is before the Lord and make atonement for it. He shall take some of the bull’s blood and some of the goat’s blood and put it on all the horns of the altar. He shall sprinkle some of the blood on it with his finger seven times to cleanse it and to consecrate it from the uncleanness of the Israelites." (NIV)
So the idea here is that the priest sacrifices an animal and sprinkles the blood on the people or the worship elements such as the ark of the covenant. The death of the animal was a reminder to Israel that there can be no forgiveness of sin without the shedding of blood. And the sprinkling is a balancing picture showing the possibility of vicarious sacrifice. Blood must be shed, but I can be made clean by the death of another.
Even in the Old Testament the people of Israel understood that these sacrifices didn't by themselves bring about true healing and forgiveness. At best they were an outward picture of the inward spiritual renewal that only God could bring about. And so the suffering servant of Isaiah 53 is one example of Old Testament writings that predict God doing precisely that: carrying out the true salvation to which the sacrificial system is only a symbol and a foreshadowing.
And then Jesus comes, and through his life, death and resurrection convinces his followers that he is that Messiah, that suffering servant, that one through whom God is truly saving the world. This is the essence of all the New Testament writings. They tell the story of Jesus, but time and again they tell it in the language of the Old Testament, because they believe that the Old Testament in all its diversity was leading up to Jesus.
So when Paul writes this in Romans 15.21...
“Those who were not told about him will see,
and those who have not heard will understand.”
... he is certainly quoting from Isaiah 52:12-15:
See, my servant will act wisely;
he will be raised and lifted up and highly exalted...
so he will sprinkle many nations,
and kings will shut their mouths because of him.
For what they were not told, they will see,
and what they have not heard, they will understand.
But he is not quoting from it in the sense that the whole prophecy of Isaiah refers to him rather than Jesus. His interpretation of Isaiah is that the death of the suffering servant will bring about healing to "many nations." This will only happen because the gospel is preached throughout those nations. So Paul's reading of Isaiah is that it is a prophecy not just of the suffering servant's sacrifice, but also the spreading of that sacrifice to the nations. Paul sees his ministry to the Gentiles as part of that process. This is why he can describe himself in these words:
"[God appointed me to be] a minister of Christ Jesus to the Gentiles. He gave me the priestly duty of proclaiming the gospel of God, so that the Gentiles might become an offering acceptable to God, sanctified by the Holy Spirit." (Romans 15.16 NIV)
It's worth noting that this is not just a proof text reading of a single verse. Both Isaiah and Paul have this theme throughout their writings. At one end, Isaiah has other Servant prophecies that point to the salvation of the nations beyond Israel. Isaiah 49.1-7 is a good example:
“It is too small a thing for you to be my servant
to restore the tribes of Jacob
and bring back those of Israel I have kept.
I will also make you a light for the Gentiles,
that my salvation may reach to the ends of the earth.” (v6)
At the other end, Paul describes in several of his letters his duty to preach the "mystery of the gospel", most notably in Ephesians 3. For Paul the mystery is that God's hidden plan of salvation is now revealed in Christ, and the church is called to proclaim that glorious story to the nations.
