Expert mode will unlock after beating the game once. it contains a harder version of the bosses with more health, faster projectiles and different attack combinations or mechanics sometimes.
However, it's not much different than regular, and since you already have experience with all the bosses, this may take less time to complete than playing on regular for the first time. You just need to defeat the bosses on expert so don't worry about your Rank.

You can check which bosses you completed on expert by pressing  then go to List, a Star mark will appear if you defeated a boss on expert. Here is a photo to demonstrate that:
Check  Sheriff, Boss,  Mayor, and  King for more information on each boss on every Isle.