Paul yearns to get to Rome — probably because he knew how important Rome was. His yearning manifests itself in his unceasing, constant prayers for the church at Rome.
That City of Seven Hills where the world’s fate was decided on Capitoline hill and where decisions were forged in the emperor’s palace across the flat plain of idol worship and where the emperor could come to a massive arched door and look down and see the Roman forum. That is the City of Rome that Paul knew was fermenting with a small clutch of young Christians. And for them he was constantly in prayer.
Paul’s prayer life, like that of Jesus and all Jews, was dotted by a routine of praying three times a day: involving the Shema (Deut 6:4-5 and maybe even more than that), perhaps the Ten Commandments, and an early version of the Amidah prayer, and I believe there is evidence in Romans itself for Paul understanding the Shema in terms of the Jesus Creed (cf. Rom 12:19; 13:9).
Whatever we know in detail, we know this: when Paul says he is praying unceasingly, he is referring to the habit of what we now sometimes call “praying the hours” or “offices” (see Praying with the Church). He stopped whatever he was doing three times a day and prayed as a matter of habit, and he prayed for Rome when he did. This is what he means when he says he prays “constantly.” (Not that I’d deny the idea that Paul was always in prayerful disposition.)
He yearned to get to Rome because of his calling: he was called to preach the gospel about Jesus Christ to all, and that meant (by definition) that he simply had to get to Rome. He couldn’t stand the thought of not getting a chance to work with the Roman church to further the gospel in Rome.
I’m glad he did.