Can I try?
Obviously, that pocket sized New Testament will be re-read many times and will be a great assistance in my journey.
One can, in absence of the Sacraments of the Church be sanctified through the reading of the scripture, as well as with the Sacraments. One thing does not exclude the other. Our spiritual work in not complete till we learn to read and love the Holy Scripture.
But the essential part in this scenario that you presented is prayer. The first prayer should be a prayer of contrition for every unconfessed sin. Then one should ask God to not tempt him with more sin. Then, given the extrraordonary circumstance, one will die in a state of grace without the sacraments of the Church.
Another thing to keep in mind is that we never pray alone, even on the desert island. We pray, always, with the entire Church in Heaven: the Communion of Saints known and unknown.
I think our challenges are greater where we are than in a desert island. If one searches for salvation, he will do well to find himself in a desert. Most of the treasure of the Church was build in a desert, sometimes literal and often deliberate desert of monastic life. Happy is he whose circumstances allow him to become a hermit.
St Jerome in the Desert
Joachim Patenier
c. 1520
Oil on wood, 78 x 137 cm
Musée du Louvre, Paris
Well, first, let me just say that I hope you don't think this is some kind of "gotcha!" question, from which there is no escape. You have cherry-picked the terms, leaving only a New Testament to read. There are a host of other "only one thing" scenarios that would not skew things in favor of your own editorial POV, but let's take what you have.
If I were left on a desert island with no chance of leaving it, and all I had to read was a New Testament, of course I would be reading it over and over, for major portions of each day! Why would I not? Do you suppose that I do not read the New (or Old) Testament at all now?
But, as a Catholic, there would be more for me to do. I would first of all find time each day to implore God's mercy, that He would give me the grace to be perfectly sorry for my sins (in other words, sorry for them solely because they offend an all-good, all-holy, all-loving God, and not because of any self-interested fear of punishment for them. I would do that because, absent the ability to obtain forgiveness through the Sacrament of Penance, I know I would have no real chance of salvation otherwise. And I know that my purpose in life is to spend eternity with God in Heaven, and it is not inherently "selfish" to pursue that goal, since it is ultimately only the pursuit of cooperation with God's positive Will. I would, then, hope for the grace to develop a totally pure and selfless relationship with my God, who, apparently, in this scenario, I will ultimately meet unshriven.
Flowing from that grace, or as a result of additional grace that God may condescend to grant to me, I would cultivate a much stronger prayer life than I currently pursue. I would hope, over time, to be able to wrap my arms around a contemplative state of prayer that I could only dream of now, surrounded as I am by the mundane cares and distractions of day-to-day life in the world. To develop such a state of prayer would be a foretaste of Heaven itself, even while imprisoned on this island your scenario places me on.
I would offer my life, and my fate on this island, in reparation for my own sins, the sins of those whom I knew in life, and the ongoing sins of the wider world around me, even those sins I will never directly know about in this life due to my being utterly cut off from them. "The prayer of a righteous man has great power in its effects" (James 5:16). I would pray that God would so construct me, through His grace, to be worthy of that verse!
So I would hope to submerge my being into the very Will of God, beg forgiveness of my sins, and hope that God would grant me the graces to walk with Him in such a way that I could intercede with Him, along with the very saints and angels in Heaven, for the furtherance of His Kingdom among my fellow men, whom I will never see again.
In other words, I would embrace my trials and death alone on this island as a vocation to become a living holocaust, submerging it into the One True Holocaust which Jesus was for us all. In this, my life on this island would come to a "purpose" far beyond anything I could likely embrace in my current, "normal" circumstances.
And, yes, a continued and ever-deepening reading of the New Testament would be a large part of my life in the scenario you suggest. But it would be a part of a greater whole, certainly. Were I left with nothing but a Tridentine Missal, I could offer that to God and achieve a similar end. For it's all about God's grace, not necessarily what I have available in tangible things. I could do as well without any printed word, provided God might be so inclined to enlighten my mind in some of the other ways I have suggested. Do you not suppose that my memory of Scripture and other pious reading would be enhanced by Him over time?