The Greek in Acts 24:18 that is translated as “rite of purification” or similar in English is translated in the German New Testament translation by Berger / Nord (publ. 1999) with den Vorschriften des Nasiräats gehorchend und geheiligt or “obeying the rules of the Naziriteate to become sanctified.”
See also complete verse (Acts 21:26) and under a vow.