You probably need the input pins with interrupt to wake up the Electron and record the button press event. Something like less than 5 seconds i would presume.
With the original LiPo one month or a bit more should be possible with only waking once an hour, but data consumption will possibly become an issue when requiring the Particle cloud each time (6KB per wake).
To keep the cloud from not requiring that extra data you’d need to wake about three times an hour, cutting into your battery life.
Also the amount of expected button presses will eat into your battery life.
If you need more battery life (up to a year) or expect many button presses you’d need to add some way to recharge (e.g. solar panel).