Except for a very small stepper a h-bridge is not a good choice - perhaps not even then. A proper stepper motor driver board (the Pololu A4988 is typical) is much more suitable. All of the complications of the order of energizing the stepper coils are handled by the driver, not the Arduino. You only need two connections to the Arduino - Step and Direction.
The problem is not knowing the appropriate current draw of the motor. The A4988 driver can supply about 1 amp comfortably and maybe a bit more with external cooling. If you have a motor that needs more amps you will need a more commercial driver which will cost considerably more but is essentially the same as far as the Arduino control is concerned.
The A4988 driver board (and other similar products) has a preset potentiometer that allows you to set the maximum current to protect the motor. If you bought one of those boards you could try your motor with a low current first and then raise the current bit by bit while checking the temperature of the motor. Expect stepper motors to run normally at a temperature that is uncomfortable to touch.
It's hard to know whether it is more cost effective to continue with the free motor or to buy a new motor and appropriate driver board that you know is suitable for your application.
Have you measured the resistance in the motor coils?
...R