There is no easy answer to this question. If there is anticipated landfall of a Category 3 or larger hurricane, it is likely government officials will urge a mandatory evacuation 24 hrs in advance of anticipated landfall.
Even "Category 2" hurricanes can be very large & cause major destruction (consider what Ike did to Galveston in 2008).
If you stay, and your property is relatively unaffected (e.g. as was the case for most of the New Orleans Metro area after Gustav in 2008), your home or apartment could be without electricity & drinkable water for a week or more. Pumping stations that depend on electricity may not be able to pump sewage. Would you want to live in 90 degree heat with no air conditioning & non-functional bathrooms under these conditions?
If you decide to stay, consider having a disaster supplies kit including plenty of food, water, medications & supplies ahead of time.
Gas stations & grocery stores will likely not be open (even if there is power) if their workers have evacuated.
Consider your Evacuation Alternatives:
Think about your best option for evacuation (Amtrack, car, bus, flights out of New Orleans airport).
Have a plan in place several days in advance of anticipated landfall.
Things to do before Evacuating by Car:
Have a plan on where you will go (e.g. a reservation in a hotel in a town 300+ miles North).
Make sure you have a full tank of gas days in advance. (Everyone else will be filling their tanks as well.)
Leave at least 24 hrs prior to anticipated landfall, and very early in the morning (if possible) to avoid the heaviest traffic (or gridlock). If you are traveling with pets, consider the time your hotel room will be available, and plan accordingly.
Listen to news radio for updated information (e.g. WWL on 870 AM or 105.3 FM).
Understand how Contraflow works, and which direction you plan to travel.
For example Westbound lanes of I-10 going from Clearview towards Baton Rouge will be diverted before reaching LaPlace and forced to head North on I-55. You will not be allowed to exit until you get past the Mississippi state line (hence the need for a full tank of gas).