I have a syringe-tube of thermal paste by Cooler Master I used a while ago (6 months ago?).
It's been lying around in my desk. Should I use this TP, or should I use the TP in the little packet that came with my CPU fan/heatsink?
As long as it is not too dried out, sure it will be fine.
I don't think that thermal paste spoils, turns rotten, or anything like that.
But you do want it the right consistancy so it spreads correctly between the 2 surfaces. If it has dried too much, it won't spread thinly, fill the cracks and imperfections as well, and simply won't conduct heat as well as it should.