I don't hold out much hope, but a service, public or private, from MSY to downtown would be amazing. I bet 80% of MSY arrivals are tourism going to the CBD or French Quarter, maybe cruise ships. The taxi line and traffic to downtown on Friday is atrocious. There is also plenty of space on existing ROW to operate such service.
There are a lot of options, but the best would probably be a light rail that follows CN/KCS from the airport, stops at Causeway/Metairie, Carrolton Avenue (former IC station), Superdome/NOUPT/Smoothie King, then a loop (mostly already built) down Howard to the convention center and cruise terminal, along the river to Canal, up canal to Loyola, back to NOUPT on Loyola.
The new Acela: It's not Aveliable.