File Geodatabase SQL is based mostly on the SQL92 standard which does not include FROM in an a UPDATE. Normally this sort of update would be performed using a subquery.
The 1.0-1.3 versions of the API are based on File Geodatabase as it existed at 10.0. This did not include support for aliasing. So "SET Field1=a.Field2 FROM Table2 a , Table1 b where a.OID=b.OID" would return an error in any case. The next release which is expected later in 2013 will be updated to 10.2 File Geodatabase SQL.
File Geodatabase support for subqueries is limited, so I can not promise that it would work in this case. I'll take a look and report my findings.