Abilities » Emerge | Abilities » Pupate Can I use Caterpie's "Pupate" Ability or Metapod's "Emerge" Ability on the first turn of the game or the turn that the Pokemon came into play? Yes. There is no restriction on the Ability. Source: Unbroken Bonds FAQ (2019-04-19), TPCi Rules Team (2019-05-02)
Abilities » Emerge | Abilities » Pupate If I use Caterpie's "Pupate" Poke-POWER to successfully evolve into Metapod, can I then immediately use Metapod's "Emerge" Poke-POWER to attempt to evolve into Butterfree, all on the same turn? Yes, you can. Source: PUI Rules Team (2008-02-21)