To "revert" New Super Luigi U's gameplay/physics engine to the one originally featured in New Super Mario Bros. U, complete the main game once. Now you should see special "M Blocks" at the beginning of every level which will temporarily change Luigi's movements to be more like his brother Mario from NSMBU (lower and shorter jump but less slipping afterward) when hit.
To play as the new character Nabbit instead of Luigi in single-player mode, simply hold down the B button on the Wiimote or the ZL button on the Wii U Gamepad or Classic Controller Pro when choosing any stage. Note that you will be in control of Luigi again after completing the stage (or if you lose a life during the stage while playing as Nabbit), so you must do this again when selecting other stages on the map.
Note that you cannot play as Nabbit when trying to catch Nabbit (obviously) or tackling one of the special enemy battles on the world map.
Use the playable Nabbit code (see "Replace Luigi with Nabbit in most single-player stages") on the final level in Star Road to play as your Mii instead of Nabbit.