When looking at a new CPU's compatibility to a system, check whether:-
1) Your CPU supports the socket that your motherboad has.
2) Your motherboard's BIOS supports your CPU, if it doesn't, check the manufacturer's website for updates that'll allow support.
3) Your current cooling setup (Heat sink and fan) supports the new CPU.
4) Your Power Supply Unit (PSU) is powerful enough to handle the CPU. This is what you meant in your above.
Fill up your current system specs and tell us which CPU it is you intend to upgrade to so that we can advise you further.
http://www.sevenforums.com/tutorials/180324-system-info-see-your-system-specs.html
Hope that helps!
Looking at those links you've provided, I reckon it should work.
The Celeron is a low end CPU that uses socket LGA 775 which your motherboard has.
Your power supply should be sufficient for the CPU.
According to that same link you gave me, support for that CPU's family was mentioned specifically, which would in turn, imply native BIOS support for that CPU.

Hello again Drake,
Here's what you're looking for in that list,
View attachment 220585
Like I've mentioned in the other thread, if the CPU support list shows your CPU, it definitely, without a doubt, means that it's 100% compatible with it, BIOS included.
A CPU would not be on a motherboard's support list by the manufacturer unless it was tested to work on that specific motherboard and it's BIOS firmware.
EDIT:- In case you're still not convinced, BIOS updates can be downloaded for your motherboard here..
ASRock > Products > Motherboard > P5B-DE > Download