Lewis wrote on a wide range of topics, but I’m unaware of anything specific to the new year that I can confirm came from him. However, the first two selections from The Business of Heaven contain some excellent thoughts that are somewhat related (when you think of wanting to make improvements).
The January 1st selection contains an except from a letter written on January 20, 1942 to Mrs. Neylan (that is in The Collected Letters of C.S. Lewis, Volume 2). He says to her that an awareness of shortcomings should actually encourage us (because we care it shows we want to be better):
“I know all about the despair of overcoming chronic temptations…It is when we notice the dirt that God is most present in us: it is the very sign of His presence.”
Then for January 2nd, the selection comes from Book IV, Chapter 8 of Mere Christianity (the chapter “Is Christianity Hard or Easy” from Beyond Personality):
“The real problem of the Christian life comes where people do not usually look for it. It comes the very moment you wake up each morning. All your wishes and hopes for the day rush at you like wild animals. And the first job each morning consists simply in shoving them all back, in listening to that other voice, taking that other point of view, letting that other larger, stronger, quieter life come flowing in. And so on, all day.”