If child is 22 years old and there is a chance that she belongs to another man other than the listed father on birth certificate can the listed father get back child support from mother?

If you live in the US, and assuming this a case where he actually paid Mom child support... Many men have tried this, and to the best of my knowledge they have all failed to be successful. There are many considerations behind that 'reasoning' by the court, but the bottom line is, you'd likely just be throwing your money away trying. But go talk to a local attorney about it.