Note that if \(D_a:=\{ p\in {Spec}(A)|a\notin {p}\} \) is a principal open subset of \(Spec(A)\), where \(a\) is an element of \(A\), then
\(Spec(h)^{-1}(D_a)\)
\(=\{ p\in {Spec}(B)|Spec(h)(p)\in {D_a}\} \)
\(=\{ p\in {Spec}(B)|a\notin {h}^{-1}(p)\} \)
\(=\{ p\in {Spec}(B)|h(a)\notin {p}\} \)
\(=D_{h(a)}\).

Now as \(O\) is quasi-compact and open, we can find a finite collection \(\{ D_{a_1},...,D_{a_n}\} \) of principal open subsets of \(Spec(A)\) such that \(O=\cup _{i=1}^n{D_{a_i}}\). Then
\(Spec(h)^{-1}(O)\)
\(=Spec(h)^{-1}(\cup _{i=1}^n{D_{a_i}})\)
\(=\cup _{i=1}^n{Spec(h)^{-1}(D_{a_i})}\)
\(=\cup _{i=1}^n{D_{h(a_i)}}\),
which is a union of finitely many quasi-compact sets and so is itself quasi-compact (it is well-known that the principal open subsets of a prime spectrum is quasi-compact).