If you cycle Yidaro but it isn't in your graveyard as the triggered ability resolves, you can't put it onto the battlefield or shuffle it into your library. You'll still shuffle your library if you haven't cycled Yidaro four times this game.

Triggered abilities from cycling a card and the cycling ability itself aren't spells. Effects that interact with spells (such as that of Cancel) won't affect them.

Some cards with cycling have an ability that triggers when you cycle them, and some cards have an ability that triggers whenever you cycle any card. These triggered abilities resolve before you draw from the cycling ability.

You can cycle a card even if it has a triggered ability from cycling that won't have a legal target. This is because the cycling ability and the triggered ability are separate. This also means that if either ability is countered (with Disallow, for example), the other ability will still resolve.

You don't have to cycle the same physical card four times for Yidaro to get tired of being cycled and stomp onto the battlefield. Any cards named Yidaro, Wandering Monster will count.