"Year of our arrival in Bordeaux"
Translation:Année de notre arrivée à Bordeaux
An is not acceptable. The different rules for the use of these two words are quite involved, but in this case the clue is the word "de". If it is "year of", it is always "année de". I think this is a case where it is impossible to find a simple set of rules that works for everything. If you use the two words in context often enough, they just start to sound right. The only thing is, I think DL has made an error. I think the french would always say "l'année" with the article.
Here is a good explanation of the different situations to use which word:
You use "notre" and "votre" when you are talking of a single object which is "owned": notre maison = our house votre ami = your friend
You use "nos" and "vos" when the objects owned are more than one (in other words, in presence of a plural word): nos amis = our friends vos voitures = your cars