"Asian" used alone, usually means rimming, that is, the licking of your anus by the woman.

There is also the Asian cowgirl  (both forward and reverse) where the gal sits upright while you are inside her, often lifting her thighs and knees up off the bed and pumping away up and down.

The former doesn't often lead to ejaculation as there is no contact (usually) with the penis while it is occuring.  It's sort of an appetizer, as it were.

The later, however, can certainly lead to you coming so that is what I would think is meant here since the term "finish" implies you will come that way.

you get fortune cookies at the end.

BBxxx