"Stand by the grey stone when the thrush knocks," read Elrond, "and the setting sun with the last light of Durin's Day will shine upon the key-hole." "Durin, Durin!" said Thorin. "He was the father of the fathers of the eldest race of Dwarves, the Longbeards, and my first ancestor: I am his heir." "Then what is Durin's Day?" asked Elrond. "The first day of the dwarves' New Year," said Thorin, "is as all should know the first, day of the last moon of Autumn on the threshold of Winter. We still call it Durin's Day when the last moon of Autumn and the sun are in the sky together. But this will not help us much, I fear, for it passes our skill in these days to guess when such a time will come again."
In our calendar Autumn ends with the Winter Solstice, which now falls on about December 20 to 22 in different years according to our present Gregorian Calendar.
If the Dwarve's New Year is the first day of the last moon of Autumn, it is the first day of the lunar cycle that begins less than 29.53 days before the start of winter in the dwarves's calendar. The Dwarves could probably still calculate and predict the date of their New Year by predicting the date when the beginning of a new "moon" or lunar cycle would fall within those 29.53 days each year. In real life people have been doing that sort of calculation for millennia before the fictional date of The Hobbit.
In European culture a "new moon" is usually the invisible all dark moon or barely visible crescent moon in the sky right beside the sun - sometimes the new moon passes in front of the sun and eclipses it. But that lunar phase would always be in the sky right beside the sun every time. Thus if the Dwarves could not see every new moon they should have used one of the other three chief phases of the moon; a full moon or a waxing half moon or a waning half moon, to be the start of their "moon" or lunar phase cycle.
In the chapter "On the Doorstep" Bilbo in a steep depression on the west side of the mountain notices the sun and moon in the sky together:
Soon he saw the orange ball of the sun sinking towards the level of his eyes. He went to the opening and there pale and faint was a thin new moon above the rim of Earth.
Therefore the Dwarves used the new moon as the beginning of their month. And since the new moon is often totally invisible to the naked eye when passing by the sun in the day sky, dwarf calendar keepers would calculate the day and time of day when the moon would be new from observing the dates of the latest visible waning crescent moon and the earliest visible waxing crescent moon in lunar cycle after lunar cycle, and thus learning to predict the date of the new moon.
Obviously the new moon would be very close to the sun at the moment of exact new moon. Because of the length of a day and the length of a lunar phase cycle of 29.53 days, the sun appears to travel about 15 degrees of arc in an hour and 0.25 degrees of arc in a minute, while the moon appears to travel about 0.5079 degrees of arc per hour and 0.008466 degrees of arc per minute.
So during the day of new moon the sun would appear to travel faster and catch up with the waning crescent moon - which would get thinner and thinner and disappear - and then pass the new moon - which would reappear as a very thin waxing crescent moon that would get fatter and fatter as the angle between it and the sun increased.
Whenever this happens half the Earth is facing the sun and moon during daytime and can observe the event with the right instruments, and half the world is facing away from the sun and moon during nighttime and cannot observe it. And it is easy for astronomers with the right tables of data to calculate and predict what time of day or night the next new moon will happen at any particular location on Earth.
If one knows what angle of separation between the sun and the waxing crescent moon is sufficient and necessary to detect the waxing crescent moon with the naked eye after the new moon, one can use tables and calculations to predict whether that waxing crescent moon will be visible to the naked eye at a particular place on the day of the new moon right before sunset.
Apparently the Dwarves of Durin's Race lost that knowledge during the fall of Erebor. Possibly there were elves in Rivendell who might have calculated the date of the next Durin's day, but there is no mention of that possibility in The Hobbit.
One way to test any theory that The Hobbit and The Lord of the Rings happen in particular past years is to calculate in which years the last new moon of Autumn (Autumn as defined in the calendars of The Lord of the Rings) will be visible in the sky right before the sun sets at about the latitude of Oxford and between about the longitudes of Greenland and the Ural Mountains.