Electronic – Not use current limiting resistor for switching device on off using TIP122 possible

Your 1W resistor exceeds the power requirement so that's perfectly fine. The only time you run into problems is if you use a resistor with a lower power rating. For example if you used a 0.25W rated resistor it may not be able to dissipate enough heat, get too hot and fail.

On the resistance side 470 ohms will cause a little less current to flow than the theoretical value of 460 ohms you've calculated, so other than delivering a little under 25mA that's fine as well. The approximately 0.5mA lower current won't make a visible difference.

If you pull the ADJ pin to ground with a transistor you'll reduce the output voltage to 1.25V. If that's good enough, all you need is a transistor like a 2N2222 with a base resistor (note that the base will draw current to turn the regulator off). It will still draw current, but it should be reduced quite a bit (the regulator with resistors will draw at least 5mA itself and the MODEM will probably draw some current at 1.25V).

If you want to shut it down entirely you could use a MOSFET high-side switch, which would involve a P-channel MOSFET and a transistor (BJT or MOSFET) to level shift the logic signal. It would not draw current other than leakage when off. If two transistors, a few resistors and a zener sounds too complicated for you, you could use a MOSFET SSR. The input is an LED so you would need to add a resistor to the output of your micro to limit the current. It should be rated for the peak current of your MOEM.

However, at 100-300mA continuous, I suggest you bin the LM317 and use a switching regulator with an ON/OFF pin. That would reduce the OFF current to tens of uA or better, and the efficiency when operating would be greatly increased. The regulator (such as LM2576) will typically have a logic-level compatible control pin so you can connect it directly to the micro.

There also exist linear regulators with ON/OFF inputs if you'd prefer.