Toward the end of most of the New Testament epistles is a section in which Paul or other authors urge their audience to do some specific things to enhance their walk with God. By personalizing these admonitions, we can have a ready-made list of resolutions for the New Year. Thus, with a little paraphrasing one can come up with the following examples:
I resolve to avoid those of my acquaintances who love to cause dissensions. (Romans 16:17)
I will be wise in what is good and guileless in what is evil. (Romans 16:19)
In the coming year, I will keep alert, stand firm in my faith, and be strong and courageous. (I Corinthians 16:13)
All that I do will be done in love. (I Corinthians 16:14)
I will put myself at the service of those people who serve others unselfishly. (I Corinthians 16:15-16)
I resolve to examine myself to see whether I am truly living in the faith. (II Corinthians 13:5)
I will agree with my fellow Christians and live in peace with them. (II Corinthians 13:11)
I will attempt to restore to fellowship those in the faith who have backslid, but to do so only in a spirit of gentleness. (Galatians 6:1)
I resolve to help bear the burdens of others who are in need. (Galatians 6:2)
I will pray faithfully for all believers, especially pastors and missionaries. (Ephesians 6:18-20)
I resolve to keep my mind focused only on those things which are pure and worthy. (Philippians 4:8)
I will devote myself to prayer, including thanksgiving for my blessings and supplications for those who have devoted their lives to spreading God's word to others. (Colossians 4:2-4)
I resolve to conduct myself wisely toward non-believers and be prepared to respond to their spiritual questions. (Colossians 4:5-6)
I will continue growing in my faith and conduct and not merely remain content with my present state of spiritual maturity. (I Thessalonians 4:1)
I will set a good example in the eyes of unbelievers in terms of maintaining a good work ethic and not making it a habit of going around criticizing others. (I Thessalonians 4:11-12)
In the coming year, I will ask God to direct my mind and heart more and more to His love toward me as shown by the giving of His Son for my behalf and the behalf of others. (II Thessalonians 3:5)
I think you get the idea from these examples, but you might personally try the same exercise on all the New Testament letters to see how many more potential resolutions you can find, and more importantly, keep.