There is no good or bad time to look for apartments in New Orleans. The city is large enough that availability tends to be constant throughout the year. If anything, you might have slightly better luck during the summer months, but it is best not to count on it. Give yourself a good two to three months to conduct your search at any time of year.
When you are applying for an apartment in New Orleans, you have to do your homework first. The after effects of Katrina are still felt in the city. Some apartment buildings that sustained water damage were never properly repaired and may have issue with moisture or mold. Always ask when a building was constructed and if it has sustained any damage before you move in.
Unless you're sticking to the suburbs, try to avoid driving in New Orleans. Parking and driving are both frustrating in the interior. On the other hand, there are streetcars and busses that provide excellent transportation throughout the most interesting parts of the city, especially the French Quarter. Transit passes are extremely affordable, with a day pass being available for $3.
Some people spend a lifetime in New Orleans and still don't experience everything it has to offer. The architecture in the city, the historic parks, and of course the music are part of what makes New Orleans so special. For more traditional entertainment, checking out a New Orleans Saints football game is fun during the NFL season. New Orleans are also famous for its festivals, so book your holidays to coincide with Mardis Gras or the Jazz festival to see the best of what the city can offer.