I fear not. It's a 787-8 route and so there is no First cabin. There is on the 787-9 aircraft, and several rumours suggest that MSY will move to that configuration, as happened to Austin (at least some of the time). I'd expect any change there to be after 6 months or so. Moreover, upgrades on BA are not a way of life, they happen, but usually it's BA's own status flyers who have a good chance on them. On the other hand, a new-ish flyer to BA may persuade the computer, known as DUT, to give you a taste of First if it sees value in doing so (e.g. because you would buy it in the future).

More detail about this, and a video talking about how well the route is doing already, towards the end of this thread:

http://www.flyertalk.com/forum/briti...gras-nola.html