If you're a Windows user, as a matter of dynamic analysis, you could try using an emulator like BlueStacks, then inspect your system network traffic with a program like Fiddler or WireShark.
Additionally, you could use a program like Cheat Engine to open the BlueStacks process and scan its memory for strings related to URLs. As an aside, you could statically analyze the APK by unpacking/decompressing it, then comb through its files looking for relevant strings.
Back to Cheat Engine, you could ultimately write a script in Assembly or Lua to automate certain tasks--or you could just change certain values in memory if there is weak server validation (or none at all).
Perhaps a significantly easier solution all around is to use something like AutoIt, which would allow you to script mouse movements and keystrokes. At that rate, all you have to do is install the game in the emulator, then script whatever process(es) you need to script for automation.