TL;DR: rather dubious claim, probably a reference to recent Putin's speech.
After a little googling the original quote is:
Вот и заканчивается проклятый 1916 год. Надеюсь, что в 1917 будет полегче.
So the cursed year of 1916 comes to an end. I hope things will be easier in 1917.
The oldest (Dec 31 2016) occurence I could find is from user
arye_11 in the comment feed of opposition-leaning Ekho Moskvy radio station. It's been removed from the original site since then, but you can still find it in Google cache. The commenter left no further hints to trace the source (such as date to look inside the diary). It is quoted as a conclusion of an anecdote from Tsar's life (original spelling):
Современники пишут, у Государя Николая Александровича было хобби, которому он предавался с упоением всей своей души.
Любил Император Российский стрелять ворон в дворцовых парках.
Стрелял хорошо, попадал часто, вел скрупулезный учет подстреленным птицам, коих общее число доходило до восьми тысяч.
Стрелял бы и дальше, да обломали удовольствие.
Вот такой "святой человек". У него государство рушилось, земля из под ног уходила, а он находил радость в отстреле птиц.
Сохранилась запись из личного дневника Николая 2-го.
"Вот и заканчивается проклятый 1916 год. Надеюсь, что в 1917 будет полегче".
Сказка ложь, да в ней намек.
Contemproraries wrote that Lord Nikolay Aleksandrovich had a hobby that dedicated him with his whole soul.
The Russian Emperor loved to shoot ravens in his palace's parks.
He shot well, hit his targets often, precisely kept track of shot birds and their number hardly reached eight thousand.
He was a master.
He would shoot even more, yet his enthusiasm has broken down.
That kind of a "holy man" he was. His country was falling apart, the land was leaving from under his feet yet he was rejoicing by shooting at birds.
A note from his personal diary survived:
"So the cursed year of 1916 comes to an end. I hope things will be easier in 1917."
The story is a lie, yet allusion is there.
So it's possible that the quote was intended as a dramatization tool from it's very beginning.
Most of the times I found the quote, it's related to last New Year address by President of Russia Vladimir Putin which started with words:
The year 2016 is coming to a close. It was a challenging year, but the difficulties we faced have brought us together and allowed us to reveal enormous resources for our movement forward.
Hence, alleged Nicolas's quote is used to suggest that also Putin's era is nigh. That was definitely intention behind Kasparov's tweet. Considering quote's timing and lack of evidence proving otherwise, I find it very likely that it's fake.