In this paper, we first give a characterization of an $f$-biharmonic curve $\gamma:(a,b)\to N^n(C)$ in an $n$-dimensional space form by solving $ODEs$. We also give complete classifications of proper $f$-biharmonic curves in a 3-dimensional sphere $S^3$ and hyperbolic space $H^3$. We also derive $f$-biharmonic Riemannain submersion equation from 3-manifolds. We then use it to study $f$-biharmonic Riemannain submersions from a 3-space form $M^3(c)$ to a 2-space form $N^2(c)$ with constant sectional curvature $c$, and also construct many proper $f$-biharmonic Riemannain submersions from $\mathbb{R}^3$, $H^3$, Sol space and a product space $S^2\backslash\{N,S\}\times\mathbb{R}$.