Power on Arduino on tilt signal
Instead of powering the Arduino off, put the MCU into power-down and have the tilt sensor generate an interrupt to wake it up. See the "Power Management and Sleep Modes" section of the MCU datasheet to see what you can use and how as an interrupt during power-down. You'll want to use avr/sleep.h and avr/interrupt.h to handle them.

Alternately, prescale the system clock down during quiet times and prescale it back up when a signal is received. See the "System Clock and Clock Options" section of the datasheet and avr/power.h for this.

In either case, shut off the subsystems you won't be using so that they don't draw power ("Power Management and Sleep Modes", avr/power.h).