I need help cat going from Seattle to Texas?!?

So I am in Seattle for my grandmas funeral and I inherited her 16 year old cat. The problems is we live in Texas and we are flying back so how do we have him on board with us? Could I take him as a carry on because on my last flight the lady behind me had a small dog in a carrier and it was just a pet no seeing eye dog or anything. So the point is this cat is old so he needs to be with us where we know he will be safe and comfy also we have never flown with an animal before so what do we do i need help!!!