Beta (β) Carinae /
MIAPLACIDUS will become the southern pole star in 6 200 AD, approaching by some 5° 42'. It is interesting to note that about the same time, Alpha and Beta Centauri will also be making their closest approach of 23'. Today, it stands as a lonely sentinel on the edges of the magnificent southern Milky Way.
From:
http://www.geocities.com/ariane1au/Page023.htm
I think there is a bright star that gets closer than that, but I can't remember which one or when.