Old psu work with new system?

If it can supply the power your new system needs, yes. It will work fine.

However, I wouldn't trust just any PSU regardless of age.

What name brand and model is it, and what type of system are we talking about powering?

Yep in terms of PSU brand is almost everything with a few exceptions at the high end.

IMHO, if your new system run single Graphic card and your old PSU have pure 500W+. I think It's gonna be ok..

Excluding the brand discussion (which is entirely valid!) ... you need to check over the connectors. Modern mobos will expect a 24 pin and 8 pin connector. Discrete GPUs will also expect 6 or 8 pin PCI-E connectors. Some of these connector types can be acquired by the use of adapter cables but honestly if you're in this kind of territory I'd seriously consider just buying a new quality PSU.