If I've had s really good striker for most of my pre-season yes, he usually does well first game. Rosenberg got a hattrick. Most new signings do well first game out if they've had a lot of practice.
I remember experimenting on CM01/02 by keeping a new striker on the bench no matter how crappy his stats. If I was one goal down or drawing a game on 70 mins I'd bring the new boy on and he'd invariably score on his debut. Probably something in the AI's personality - increased determination or something on players' debuts. Got a few "youngest ever goalscorer" records (whatever happened to that BTW?).