Short answers, pulled from the story.
The Old New Year falls on the 14th of January in the Gregorian calendar. This corresponds to the 1st of January in the Julian calendar, which is still used by some jurisdictions of the Eastern Orthodox Church.
The Old New Year falls on the 14th of January because it follows the Julian calendar, which runs thirteen days behind the Gregorian calendar in the 20th and 21st centuries. Governments across Russia and Eastern Europe once used the Julian calendar, and the Eastern Orthodox Church continues to mark its feasts by it.
Countries that observe the Old New Year include Russia, Serbia, North Macedonia, Ukraine, Belarus, Georgia, Armenia, Azerbaijan, Kazakhstan, Uzbekistan, Palestine, Jordan, Bosnia and Herzegovina, Montenegro, and Switzerland. Scotland and Wales also maintain Old New Year traditions.
In Serbia the Old New Year is commonly called the Serbian New Year, and sometimes the Orthodox New Year. Celebrations in Belgrade typically include a concert in front of City Hall or the National Parliament and fireworks fired from the Church of Saint Sava.
Hen Galan, the Welsh Old New Year, is celebrated with children going house to house singing traditional Welsh-language songs and collecting gifts of sweets and money. Major traditions include the Mari Llwyd, the making of apple charms called perllan, and the collecting of calennig, with the best-known celebrations held in Cwm Gwaun, Aberystwyth, and Cowbridge.
The Burning o' the Clavie is a fire ceremony held in Burghead, Morayshire, Scotland, on the eve of the Old New Year. In Scotland the Old New Year falls on the 12th of January rather than the 14th.
The playwright Mikhail Roshchin adapted his 1973 comedy-drama The Old New Year into a screenplay for a 1980 television film released by Mosfilm studios. The film featured music by Sergey Nikitin and poetry lyrics by Boris Pasternak.