First, the direct answer to the question is yes, you may leave the train and leave the station and go anywhere. No problem, so far as that is concerned. Be sure you take your ticket stub with you so there will be no question about your right to re-board.
But something you may not realize, living in the U.K., is that the New Orleans area has recently been extremely devasted by Hurricane Katrina, first, and Hurricane Rita, second. Let's just hope all the city and the railroads are functioning normally at the time you intend to travel. Your agent probably booked you before all this happened.
And it was not just the hurricane winds themselves---some of the levees which kept New Orleans protected from flooding broke during both storms.
And, if that is not enough, the very area the train takes out of New Orleans heading towards Jacksonville is what we call the "Gulf Coast area" and all of that is in extroardinarily bad shape from the first hurricane, Katrina.
This recent storm, Rita, specialzied in tearing up parts of Texas (as well as Louisiana) through whicih the Sunset Limited passes, making the news even more bleak. All of this has been going on about a month.
Secondly, even if all things are running smoothly by then(and who can say???) it also happens to be true that the Sunset Limited is often impacted by freight delays and can run hours late, with great ease, I am chagrined to say. Amtrak does not own these tracks and the freight railroads often put their trains ahead of the passenger trains. This means that all that layover time in New Orleans is largely a cushion for potential delays. Sure, it arrives on time sometimes. But if it arrives two or three hours late, you will see, it then has a chance to get a fresh start and roll out of New Orleans on time.
Sorry to be the bearer of such dismal news....my advice is to stay in touch with the forum, check other listings that pertain to the flood and storm interruptions of service and see what develops.
Good luck!! To you and to ALL of us who ever want to go to New Orleans and environs again.