Usually I only get soreness for a workout or two after adding an exercise I haven't done in a while.
For example, I'm running Dan John's Armor Building Formula right. It involves double kettlebell front squats, which I hadn't done in a while. My quads were sore for the first two sessions. After that, no soreness unless I really push things.
Push ups utilize biceps and triceps (to a lesser or greater extent based on the type of push up) so what you're experiencing tracks.