Dear Andrea, the fact you mention is indeed true. Consider the associated morphism of affine schemes $Spec (\phi) = f : X=Spec B \to Y=Spec A$. Your hypothesis on unit ideal generation translates into the fact that the open subsets $U_i= Spec B_{f_i}$ cover $X$. The finite generation of $B_{f_i} $ as an $A$-algebra says that the scheme $U_i$ is of finite type over $Y$. And finally the result you request, that $B$ is finitely generated over $A$, is equivalent to the geometric statement that $X$ is of finite type over $Y$. But this last assertion is true by EGA I Proposition (6.6.3), page 153.
Of course one could unpack all this and give a purely algebraic proof. But I think it is nice to see things geometrically and, in case you don't know elementary scheme theory yet, this might motivate you to learn it . At the level used here it is little more than a language in which to speak of algebra. [ By the way, don't be afraid of the hypothesis that the morphisms appearing in this context be quasi-compact: all morphisms between affine schemes are quasi-compact !]