How to provide power to Arduino after a remote button is pressed
This post is not about removing the power, but put it to sleep and wake it up via an interrupt via a signal. However, in this case, the Arduino should use only a few uA and not drain your battery. Make sure that you also power down the sensor and unused peripherals. To get rid of the last uA's, you can use the relay as is proposed in the comments.

You can set the interrupt with: 

attachInterrupt(0, wakeUpNow, LOW); // use interrupt 0 (pin 2) and run function, 


Create a function for the interrupt (wakeUpNow)

void wakeUpNow() // here the interrupt is handled after wakeup
{
   ...


And in the loop let your Arduino sleep

sleepNow(); // sleep function called here


Where sleep is a function like this:

void sleepNow() // here we put the arduino to sleep
{
 set_sleep_mode(SLEEP_MODE_PWR_DOWN); // sleep mode is set here
 sleep_enable(); // enables the sleep bit in the mcucr register
 attachInterrupt(0,wakeUpNow, RISING); // use interrupt 0 (pin 2) and run function
 sleep_mode(); // here the device is actually put to sleep!!
 sleep_disable(); // first thing after waking from sleep:
 detachInterrupt(0); // disables interrupt 0 on pin 2 so the


}

For a complete example, see Waking up an Arduino with input from a sensor