Circuit to shut off power after a short time of turning on
There is no way around the switch drawing current if you need to continuously monitor it.  However, there are two points that help:

The current can be very small.  With a 1 M&Omega; pullup driving a CMOS input, it will draw only 3 &micro;A from a 3V supply.  That is small even for some button batteries.

You don't need to continuously monitor the switch.  For most rolling ball motion switches, every 20-100 ms is good enough depending on your application.  You can have a tiny micro wake up every 20 ms.  It would turn on the pullup just long enough to take a reading, then turn it off again.  The average current drain can be below a &micro;A with this setup.  That's less than the self discharge of some button cells.