Yes, the location of the First Class car is entered into the computer at the originating station and any agent worth their salt can look that information up.

Of course one also must rely on the originating station to enter the correct location. I've had it happen twice where they got it wrong. The first time was about 2 years ago out of Boston, and I didn't care since I boarded in Boston. But at every stop down the line, people were positioned at the wrong end and either we had to wait for them to run down the platform or for the crew to get them onboard and then have them walk through the train.

The second time was just a month ago, when I boarded at Philly on NTD. We were all standing at the head end as that was where both the lounge and the usher in the main station had directed people. Of course as the train rounded the curve just south of the station, I saw that the FC car was on the rear and started the long walk to the other end of the platform.