No, the whole thing about using an Arduino, is that one device does everything. So it is just four buttons connected to four Arduino pins, a "1602" display with "backpack" and an Arduino Nano.
A UNO is not very suitable for such projects and while a Pro Mini would perform the job perfect,y, you have to connect another module to program it and you would not like that. 
Batteries are a nuisance - you have to charge them or replace them, usually at the wrong time when you go to use it and it doesn't work, and you have to design the device to shut off all power until you press one of those buttons, which complicates matters. You can plug it into your computer, but you can instead plug it into a USB "phone charger" which is still plugged into the power point (wall outlet) but you don't need the computer.
I am not sure what to advise about what buzzer to use. Do you want a high pitched one or a low pitched - or what?
Be warned that to properly build it you really need to be able to solder. 