Yes you can - you earn AA points/miles for the underlying economy fare. You would need 35,000 AsiaMiles for the oneway upgrade from LAX to HKG.
The upgrade availability in advance is the same as the business award, but CX yield management is difficult to forecast - seats may appear again at any time, although LAX is one of the routes with the highest J demand.
Amex MR is the obvious way to get miles into AsiaMiles quickly, I think, but I am not based in the US so I'm not sure.