What year was this set in?

I don't think any year is mentioned or inferred. I was always under the impression that it was very very very very far into the future. And you could just chalk the black and white picture up to artistic choice. Even now people take pictures in black and white because they like the aesthetic of it.

From imdb trivia

Ad Astra is set in 2119/20, 100 years in the future from when it was released. The following assumes Roy McBride's narration is more reliable than the recollection of events he made to others, which is not consistent. The wall plaque about Roy's father's career states that the Lima Project took off on 28 Dec 2079, and Roy says he was 16 when his father left, so he was likely born in 2063. Brad Pitt himself was born in December 1963, so it would appear that the dates were chosen to make the character be born exactly 100 years after the actor was. Roy was 29 when his father disappeared, and his final (public) message is described as arriving 27 years ago, making Roy 56 years old at the start of the film, the age Brad Pitt was in 2020. However, H. Clifford McBride is 15 years "older" than Tommy Lee Jones, making him almost 90 years old when Roy reaches Neptune