Can one reduce a problem of unknown complexity to a hard problem to show hardness? In this paper (page 3 Theorem 1) the authors want to prove that their problem is NP-complete. Their method is as follows. Let their problem be known as $P$. They show that their problem can be written as a $0\text{-}1$ integer program. Then they claim that $0\text{-}1$ integer programs are NP-complete and therefore their problem $P$ is NP-complete.

I find this proof hard to believe. For the problem $P$ to be NP-hard I think one has to reduce the $0\text{-}1$ integer program into an instance of problem $P$ and not the other way around.

Please can someone explain if this proof in the paper is acceptable?
The proof is not correct.  As you've indicated the reduction needs to go in the other direction, i.e. integer programming must be reduced to their problem, not the other way around.