Yes, the 555 timer is for that. It's not obvious to me though how you would get just a single 555 timer to count X number of times and then stop. How many 555 timers were you given? Just one? Or two?
With two, you would wire each 555 timer in a monostable configuration to transmit a delayed pulse. Each 555 timer sends a signal to the BCD counter to increase the score, but that same signal also triggers the next 555 timer. You also give each 555 timer a button that can also trigger it.
So when you push a button, it immedietely increases the score by 1. But since the 555 timer is also triggered, the 555 timer will send another pulse shortly after to increase the score by 1 again. But this pulse also triggers another 555 timer, so that 555 timer increases the score by 1 again, etc.