The problem is that the last daily Eastbound Amtrak train (from Buffalo or Rochester to Syracuse) departs fairly early in the afternoon. That train, which originates in Toronto, is sometimes late, and sometimes on time. So unless you were to arrive in Buffalo by late morning, or in Rochester by about 1:00 PM, you probably could not be assured of connecting to the last Amtrak departure to Syracuse. That said, it is possible to get from the Buffalo Airport to the (downtown) Buffalo Amtrak station at Exchange Street, and from the Rochester Airport to the Rochester Amtrak station by very cheaply, by local bus. Here are links to the local bus schedules and trip planner. In each case, just enter "Airport" as your origin, and "Amtrak" as your destination, and you will be given the necessary options. (FYI: The Buffalo Airport is in "Cheektowaga.")

Buffalo bus schedules and trip planner:
http://www.nfta.com/metro/index.asp

Rochester bus schedules and trip planner:
http://www.rgrta.org/

It is also possible to get from either the Buffalo or Rochester airports to those cities' intercity bus terminals by local bus. Intercity service is provided by Greyhound and Trailways.

Service from Rensselaer (where the Albany Amtrak station is located) to Syracuse is more evenly spaced throughout the day, but getting from the Albany Airport to the Rensselaer train station requires either three buses, or one fairly expensive taxi ride (I'm guessing about $30 or so, with the tip). So I would try to make Buffalo or Rochester work, if at all possible. From what city is your trip originating?