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)

Relevant cards

Note: All applicable cards may not be listed below.
SM10_2
SM10_2
SM10_3
SM10_3