Fortunatly, stephenthorodso was wrong. There is a way to boot CAT B15 Android phone that is turned off and power button is broken.
Here are the steps:
1) download and install "Minimal ADB and Fastboot" console application for Windows
2) Connect your phone with PC via USB (it will just show battery charging, but don't worry)
3) Start "Minimal ADB and Fastboot" from Windows Start Menu -- it will open Command Prompt
4) Take out the battery from the phone
5) Keep pressed "Volume Down" and the button between Volume Down and Up (however it is called)
6) While these two side buttons are pressed, put the battery back into phone
7) The phone will enter the "Fastboot mode" (you'll see small text at bottom-left corner of the screen)
8) Now, back at the Command Prompt
- you are probably already at fastboot installation path: "''c:\Program Files (x86)\Minimal ADB and Fastboot''" - if not, go there: ''cd c:\Program Files (x86)\Minimal ADB and Fastboot''
- now, to make sure your USB connection with the phone works well, type:
''fastboot devices'' [enter]
you should get something like: mt6582_phone fastboot
9) now type: "fastboot continue" [enter]
wait 10-15 seconds and you'll see your CAT B15 booting into normal Android!!!!
(just worked for me and I'm super happy)