A shuttle from the airport will be slower and cost more ($24/person) than a cab ($36/2 people). So cab for sure, or Uber/Lyft for about the same price.
The Staybridge is decent and centrally located.
Expect it to be hot.
There are lots of walking tours in the Quarter - my mother took the one offered in association with the state history museum and liked it a lot:
Friends of the Cabildo Book Tours | New Orleans French Quarter Walking Tours Booking
The WWII museum generally gets rave reviews.
The Historic New Orleans Collection is worth a visit.
Mardi Gras World is unique and entertaining.
There are a ton of midsize and smaller museums in town:
https://en.wikipedia.org/wiki/List_o...in_New_Orleans