Although, I found no direct remarks or a mention of what specific role did that Asian play in the first plot of Amsterdam to kill Bill The Butcher,but through these scenes, we can come to a conclusion.
I did not find any uploaded videos so created a screenshot in order.
Amsterdam goes to the Chinese Pagoda and shows this Asian person a print of the hand made picture of Amsterdam's father being killed.He points to himself telling he is the son. Then this person says nothing but lights up a shot of some alcoholic drink to which Amsterdam nods. This was enough for Amsterdam to understand that this person was referring to that celebration when Bill drinks such a shot. Its then that he should take his chance.
Amsterdam: Every year the Natives celebrated the killing of my father
all over again at Sparrow's Chinese Pagoda in Mott Street. The Chinks
hated the Natives worse than we did.The drum rolls and the Butcher
drinks a glass of fire. When you kill a king you don't stab him in the
dark. You kill him where the whole court can watch him die.
You very rightly presume that just before Amsterdam's attempt fails at the Chinese Pagoda, that Asian individual gets taken away by Bill's men. It would be a stretch to imagine what he was doing there, but for your specific question my answer would be: The main role played by that Asian was to assure Amsterdam that he should kill The Butcher at that celebration when he is about to drink that shot with fire.