Value of resistor to limit current to 1 ampere in 12 volts circuit
A 12 V source will deliver whatever current to the modem it wants to draw. There is not only no requirement for a current limiting resistor, but any resistor large enough to limit current will be positively harmful for the modem's operation.
A more serious problem is the tolerances on the voltage. Does the modem need 12 V +/- some tight tolerance which is provided by the power supply that came with it, or will it tolerate the sort of variations that you could get from a '12 V' battery? A lead-acid battery could be 14 V on a float charge, and 11 V when fully depleted.
In practice, there's a good chance that the first module in the modem is a buck converter, and will accept quite a wide variation in input voltage. How lucky do you feel?