The man, with the last names of Caicedo Lopez, was detained in the early hours of Sunday and the concert in Alajuela went ahead as planned.
The threats were written in Arabic and made online.
Security has been tightened at Grande's concerts since the attack at one of her Manchester shows.
Fans had to pass through three security checks for Sunday's gig.
Costa Rica police chief Walter Espinoza told the Costa Rica Star that even if the threat had been made as a "joke", they had "to verify whether or not there is a real threat, because this is a very sensitive situation and it could lead to a tragedy".
The 24-year-old singer posted an image from the show on her Instagram account but did not mention the threat.
She is due to play four concerts in Mexico over the next eight days before moving on to Japan in August.
Twenty-two people were killed in the Manchester Arena attack, including seven children.
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A 22-year-old Colombian man has been arrested on suspicion of threatening to attack an Ariana Grande concert in Costa Rica.