It's easier than you'd think. At New Orleans the river's average speed is only 3 mph. That means you just have to get a wind speed that beats that. Here's a diagram from the University of Pittsburgh that deals with the Allegheny and Ohio rivers, but the principle is the same.

Now, this all depends on the wind, and unfavorable wind conditions mean that it isn't an easy process by modern standards; but it can be done.

New Orleans was founded where it was by Jean-Baptiste Le Moyne de Bienvielle because it was convenient high ground adjacent to the portage between the Mississippi River and Lake Pontchartrain. That portage cut off 100 miles of river travel for cargo bound for the coast.

As for sailing upriver, you're going to want to look at this Report on the Internal Commerce of the United States from 1888. What you want starts at the bottom of page 183 and continues for about 20 pages; it's a description of how river trade functioned between 1800 and 1840.

Here's an excerpt from page 185:

"For the transportation of freight up-stream various kinds of boats were used, but none of them can be said to have proved successful, and the tonnage up was barely 10 per cent of that floating down. The system of rowing up the river and against the current was tried. It was slow, tedious, and expensive. The boats coasted along the shore so as to avoid the full force of the current, but it required one oarsman for every 3,000 pounds of freight, and the work was so tiresome that the men rested every hour.

To travel from 14 to 30 miles a day was considered very good work. The river was crossed at the lower end of each bend, and in the crossing the current carried the boat down a half a mile or so. It is said by old boatmen that they were compelled to cross the Mississippi 390 times between New Orleans and Saint Louis."