I have fond memories of the Z80 - I built my first computer around it! However, the Z80A that you have is quite a lot less powerful than the processor in the Arduino. AFAIR the Z80A is clocked at 4MHz maximum and takes 4+ clock cycles per instruction. The atmega328p in the Arduino is clocked at 16MHz (although it can go up to 20) and executes most instructions in a single cycle. So it's around 16 times faster. Also, you need to interface the Z80 with RAM (such as your MK6116N) and ROM (such as the ROM chips you have) to get it to do anything useful, whereas the atmega328p has RAM (although not very much) and flash ROM on chip. You need some TTL logic to do the interfacing. Also you need to provide a clock signal.
So, unless your aim is to reconstruct something historical, it's a lot easier to write programs for the Arduino than to try to run anything on the Z80.