Additionally, if you have the opportunity, I would recommend that you take part in Eucharistic Adoration. If you have a copy of the Liturgy of the Hours, you can take part in some of the hours, including Vespers. This may be done as I suggested here, in accord with the practice in the Vatican. As a friend pointed out, if you celebrate the office of Matins (Office of Readings) as a vigil before midnight, you can also fulfill the plenary indulgence for the Te Deum (as the Te Deum is part of that particular Office.
Right now, however, I would wish to draw your attention to some observances for the day of Jan. 1, Solemnity of Mary, Mother of God, and World Day for Peace. As a Marian Feast, it is appropriate to pray any number of Marian prayers, including the Rosary, Angelus, Alma Redemptoris Mater (Loving Mother of the Redeemer), etc. Given that this day is held in honor particularly of Mary's title as Mother of God, however, it seems worthy that one recite or sing the Akathistos hymn to the Mother of God. This is a hymn to Mary derived from the practice of the Eastern Church. While I know of no music for this hymn, there is an old practice of singing recto tono (on one single note) a liturgical text. Such could be applied in this case.
Additionally, there is a plenary indulgence for reciting or singing the hymn Veni Creator Spiritus on the first day of the year. This hymn is traditionally sung for beginnings of things, calling on the Holy Spirit before endeavoring something new. This prayer would be excellent to recite or chant before meditation on any occasion.
Finally, I would include this prayer for peace from the book Catholic Household Blessings and Prayers from the Bishops' Conference.
Immaculate Heart of Mary,
help us to conquer the menace of evil,
which so easily takes root in the hearts of the people of today,
and whose immeasurable effects
already weigh down upon our modern world
and seem to block the paths toward the future.
From famine and war, deliver us.
From nuclear war, from incalculable self-destruction, from every kind of war, deliver us.
From sins against human life from its beginning, deliver us.
From hatred and from the demeaning of the dignity of the children of God, deliver us.
From every kind of injustice in the life of society, both national and international, deliver us.
From readiness to trample on the commandments of God, deliver us.
From attempts to stifle in human hearts the very truth of God, deliver us.
From the loss of awareness of good and evil, deliver us.
From sins against the Holy Spirit, deliver us.
Accept, O Mother of Christ,
this cry laden with the sufferings of all individual human beings,
laden with the sufferings of whole socieities.
Help us with the power of the Holy Spirit to conquer all sin:
individual sin and the “sin of the world,”
sin in all its manifestations.
Let there be revealed once more in the history of the world
the infinite saving power of the redemption:
the power of merciful love.
May it put a stop to evil.
May it transform consciences.
May your Immaculate Heart reveal for all the light of hope.
You may undertake to pray any or all of these prayers. It may sometimes be difficult to pray. We should take advantage of the prayers which have been provided by our tradition (as well as prayers reflecting our present difficulties as with the prayer provided just above). New Year's Day is a time to begin again. With the help of Our Lady's prayers, let us do it.