Kulich made his professional debut for HC Energie Karlovy Vary in the Czech Extraliga during the 2020–21 season where he appeared in eight games.
[3] On 15 July 2022, Kulich embarked on his North American career after he was signed to a three-year, entry-level contract with the Sabres.
[5] Kulich represented the Czech Republic under-18 team at the 2021 IIHF World U18 Championships where he appeared in four games.
[8] He again represented the Czech Republic at the 2023 World Junior Ice Hockey Championships where he recorded seven goals and two assists in seven games and won a silver medal.
He again represented the Czech Republic at the 2024 World Junior Ice Hockey Championships where he recorded a team-leading six goals and six assists in seven games and won a bronze medal.