Up Taunt: Holds the Kokiri Sword up in the air, crying out "Hup!".
Down Taunt: Sheaths the Kokiri Sword and places the Deku Shield on his back and takes a drink of Lon Lon Milk.
Side Taunt: Places the Deku Shield on his back and watches Navi fly around him. This was originally Link's side taunt in Brawl and SSB4.
Descends onto the stage while surrounded by rays of blue light that shine down from the sky, referencing the Ocarina of Time's powers. Upon landing, he equips the Kokiri Sword and Deku Shield. This is akin to his older self's on-screen appearance in SSB.
Looks around himself.
Kicks the toes of his boots on the ground, as if knocking off the dirt.
Left: While holding his sword in a reverse grip, adjusts his belt and flicks his nose. Based on one of Young Link's idle animations in Ocarina of Time.
Up: Stabs forward with his sword, then holds it up, giving off a battle cry. This is nearly identical to one of Link's poses other than the camera angle and Link using his right hand to hold the sword, while Young Link uses his left hand.
Right: Sheathes his sword, dusts his hands off, and puts them on his hips while grinning.
In competitive play
To be added
Young Link and Pichu share the longest gap between playable appearances in the Super Smash Bros. series, with seventeen years between Melee and Ultimate.
This has been noted by the Super Smash Blog's "Today's Fighter" description for Young Link.