Upgrading from 2.53Gz to 3.0gz

I recently Acquired a 3.0gz Pentium Processor from a desktop in the trash, along with spare parts in case my desktop ever fails. I don't know if it's dual core or single yet, but being an old model computer i found from the garbage my luck isn't going to go too far. My other desktop had a 2.53gz Single Core processor in it, and seemed to have a good sized heat sync, but i noticed it did not have A TRACE of Thermal paste in it. I will go out and get some soon to put on, but I can't even use the desktop yet because I don't have a cord for the monitor (I've used it before, it has XP on it and runs decent) It has 1gb of ram (I upgraded it when i first got it) and I found another gig of ram in the "Trash" computer(now an empty case in the garbage).

I may upgrade the Power Supply when i check it out to one with a bigger fan (dunno if it will help) and I think the actual fan's decent sized. The Processor fit perfectly into place in the desktop, I'm just not sure if it's a good idea to run it like that, overheating is my main concern. Should i be fine?

yep, I'll get some as soon as possible, I really have no reason to boot it up since i have no monitor, so that'll be the first thing i do, and, my CPU will be sitting on my desk, it could be useful one day considering i see this model computer alot :)