Hi MC and welcome to the Ford SYNC Forum. No...A SYNC software version update (i.e. 3.0v20204) is not the same as an APIM firmware update. A SYNC 3 software update would not address an APIM reprogramming and/or rear facing camera issue (assuming that is what you mean by "cam").
You either need an APIM update or an APIM replacement. However, that is difficult to diagnose over the Internet.
As to the labor time? I have no guess why they are telling you 1-3 hours. Don't know if that is the "book" time or they are guessing.
Keep us updated and good luck.