lol you guys were certainly talking about when you guys were born, not literally "age".. When I first read what you wrote I was like what the hack is going on hahaha
Anyway, yes, I think she meant exactly that and it seems that her younger sibling was born in 1999. (동생 could be either brother or sister but certainly younger one)
For the question part, I think you can simply say "편하게 언니라고 불러도 돼(요)?" which means "can I call you just (comfortably) unnie?" You might think calling her "unnie" is formal but it is not necessarily so. It feels very natural to Koreans to call a friend who is slightly older "형" "누나" "언니" and your friend might feel very comfortable talking to you.