Measuring the rotation angle of stepper motor
Yes, just keep track of the number of steps you have given the stepper driver. Usually a counter is used in the program. On power-up you would rotate the arm until a 'home' switch is made, zero your counter and then keep track from then on. It's a very simple and economical method and is used in thousands of applications including, for example, dot-matrix and inkjet printers.

Stepper control is generally 'open loop'. This means that if the arm in your application gets blocked or stalled the program position counter will not know and will be out of sync with the real position.

Step angle will be determined by your motor and whether you're using full-step, half-step or some micro-step mode.