The other direction is well known I think it is true and I was told by several other guys doing algebraic geometry that it is indeed true but they did not know how to prove.I am also wondering whether there is a general nonsense style proof.It looks like the statement can be proved by playing adjunctions in several categories.Note that a scheme $X$ is quasi compact iff the structure sheaf $O_X$ is compact object in category of quasi coherent sheaves $Qcoh(X)$,we also know that if $X$ is noetherian scheme,then coherent sheaves are exactly compact objects in $Qcoh(X)$,so it looks like this statement is equivalent to say if $f_*:Qcoh(X)\rightarrow Qcoh(Y)$ preseves compact objects,then $f$ itself is proper morphism of scheme.But this somehow means that $f^{-1}$ preserves compactness in the topological sense.(I think this can also be phrased into general nonsense language,say the compact objects in category of topological space).Then one might be able play the commutativity of $Hom(M,?)$ functor and filtered colimits game to get proof.

All the comments are welcome Thanks