I can see a few possible problems, but I can't say for sure.

1. The brakes may not mount the same way, the old brakes are probably nutted and the new brakes may have a recessed allen bolt. You can always modify things a bit to make them work, but it adds a bit of difficulty. Here is a picture of the difference:


2. The brake reach may be different. Meaning the actual arm of the newer brakes may not be long enough to reach the rim surface once you mount it on the old frame.

3. Depending on the old frame, the seat post from the Fuji may be the wrong size for your old frame.