You can trigger this fight by collecting at least 15 Cat Shines, which can be done as soon as you complete the first three full islands in the game. If you forgot, you will need to wait until Fury Bowser is out and about, then the Giga Bell will also be active; collect the Giga Bell like any other power-up to begin the boss fight. Fury Bowser in the second fight is almost the same, as you will only be able to damage him upon him going upside down, but he does add a new move into his rotation. Luckily, you won’t have to wait long for the new move, as he will do it at the outset of the second fight.
Bowser will go on his side and begin rolling in your direction, with him keeping the momentum and movement up for a little bit. Eventually, he will stop and starting revving his “engine” for one final attack like this, then he will stop and jump away, gearing up for a breath attack. After the breath attack will be the ground slam plus shockwave, which means you can do your Ground Pound on Bowser to damage him. You will only be able to bring Fury Bowser’s health down to around half, before he will retreat and return to his normal hiding ways. So, you might only have to deal one or two hits on Bowser, depending on when you collected Cat Shines.