Paul is no longer under house arrest when he wrote this letter to Titus. He made visits that are recorded in other writings that tell about Paul's travels to the furthermost reaches of the Roman Empire (which abruptly ended at Hadrian's Wall in Scotland). Monasteries, museums, cathedrals, and even the Vatican itself records Paul's travels into Western Europe. Paul went to other places, travels that are not recorded in the canonized Bible. In these other travels Paul established a faith in Jesus as the Messiah to the people of Western Europe. There is a link to a website that tells a bit about Paul's other travels.
http://www.matthewmcgee.org/paultime.htm

After preaching both in the east and west, he gained the illustrious reputation due to his faith, having taught righteousness to the whole world, and come to the extreme limit of the west, and suffered martyrdom under the prefects" (The First Epistle of Clement to the Corinthians, Chapter 5).

Clement lived at the same time as Paul and had knowledge of Paul and his travels.

The letter to Titus is full of warnings about false teachers and false doctrine. Titus was a Greek believer taught and nurtured by Paul, who later became the overseer of the churches on the island of Crete. This letter helps us know how to train others to lead and teach. The church there was most likely started in Jerusalem at Pentecost.

We must remember that Paul was preaching to the Gentiles and his letters were to the leaders and teachers of the Gentile churches. Most of them did not have the background of the scrolls (the Old Testament). Possibly many of them worshiped pagan gods before being introduced to Jesus. Most of the believing Jews did have this background. The Gentiles didn't have a moral code from God or even from their pagan gods. They weren't even looking for a messiah like the Jews were.

Titus 1:2
a faith and knowledge resting on the hope of eternal life, which God, who does not lie, promised before the beginning of time,
Eternal life was promised to us before the beginning of time. These plans were established before the creation, before time (as we know it) began on the fourth day of creation. What an amazing thought! But not only does God promise eternal life through Jesus, He has promised us wisdom, destined for our glory before time began.
1 Cor 2:7
No, we speak of God's secret wisdom, a wisdom that has been hidden and that God destined for our glory before time began.And not only does He promise us eternal life and wisdom and glory, He redeems us and He summons us by name!
Isa 43:1
But now, this is what the LORD says-- he who created you, O Jacob, he who formed you, O Israel: "Fear not, for I have redeemed you; I have summoned you by name; you are mine.And not only does He summon us by name, He leads us out!
John 10:3
The watchman opens the gate for him, and the sheep listen to his voice. He calls his own sheep by name and leads them out.
Totally AMAZING!

How do we “make the teaching about God our Savior attractive”? (Titus 2:10).  There is a fine line between making the teaching attractive and compromising the truth.

Titus 2:15
These, then, are the things you should teach. Encourage and rebuke with all authority. Do not let anyone despise (disregard) you.“Encourage and rebuke” is to build others up but to also hold others to account, as we should have someone to hold us to account. In other words, we need a Barnabas in our lives.

Believers should always be "...eager to do what is good.."
To be eager is to have a keen interest in something. In this case we should have a keen interest in doing good things. But, of course, it goes way beyond that. It also means that we should be burning with zeal, vehemently contending. So, not only should we have a keen interest in doing good things but we should be burning with zeal to be doing good things. That takes some effort!