I found a 15th century Medieval inscription that abbreviates to the last two numbers of the year (cf. "Summer of '69"):
HIC IACET DOMINUS WILELMUS MEDELEY ABBAS HUIUS MONASTERII XII QUI OBIIT XII DIE MENSIS DECEMBRIS ANNO [DOMINI MILLESIMO QUADRINGENTESIMO] SEPTUAGESIMO TERCIO CUIUS ANIME PROPICIETUR DEUS AMEN.
Here lies the Sir William Medeley, the 12th abbot of this monastery, who died on the 12th day of the month of December in the year '73. May God look kindly on his soul.
Otherwise, I have been unable to find a consistent abbreviation.
This should not strike us as a gross omission. Obviously, the Romans never had to speak about years like this. English is actually fairly unique when it truncates a 4-digit year into 2 2-digit numbers. Consider 1999 (I'll exclude the word for "year"):
- English: nineteen ninety-nine (5 syllables!)
- French: mille neuf cent quatre-vingt-dix-neuf (8) or dix-neuf cent quatre-vingt-dix-neuf (8)
- German: neunzehnhundert[-]neunundneunzig (8) or neunzehnneunundneunzig (6--"more colloquial")
- Spanish: mil novecientos noventa y nueve (10 syllables with Synalephas)
- Italian: millenovecento novantanove (11)
- Latin: [anno Domini] millesimo nongentesimo nonagesimo nono (16)
So sure, it's longer, but we're not talking about orders of magnitude.
As further evidence, both the Vatican and Nuntii Latini have many examples of the full form without any attempt to shorten.
It is difficult to prove a negative (so I am happy to be contradicted!), but I think I can venture to claim that there is no commonly occurring abbreviation of years in Latin.