I think without the hardware interrupt features like the X2 models have, youre sort of limited on what you can do. Have a look at this thread:
http://www.picaxeforum.co.uk/showthread.php?8353-Low-Power-Battery-Backup-Reference-Design&highlight=08m+blinking+years
Another option is to design a power circuit based on the push button. Like have the button pull the base on a NPN transistor high which would pull the base of PNP low which would power up the PICAXE. The PICAXE's first order of business would be to hold the NPN's base high. Then when the program is finished, have it put the NPN's base low which would turn off the PICAXE. Here, check out this thread, lots of pictures are better than me trying to describe it:
http://www.picaxeforum.co.uk/showthread.php?22601-circuit-help-momentary-switch-for-power-on-picaxe-to-power-off
