The fraction of $J/psi$ mesons which come from B-meson decay, $textrm{F}_{B{rightarrow}J/psi}$, is measured for J/$psi$ rapidity mbox{$1.2<|y|<2.2$} and $p_T>0$ in $p$+$p$ and Cu+Au collisions at $sqrt{s_{_{NN}}}$=200 GeV with the PHENIX detector. The extracted fraction is $textrm{F}_{B{rightarrow}J/psi}$ = 0.025 $pm$ 0.006(stat) $pm$ 0.010(syst) for $p$+$p$ collisions. For Cu+Au collisions, $textrm{F}_{B{rightarrow}J/psi}$ is 0.094 $pm$ 0.028(stat) $pm$ 0.037(syst) in the Au-going direction ($-2.2<y<-1.2$) and 0.089 $pm$ 0.026(stat) $pm$ 0.040(syst) in the Cu-going direction ($1.2<y<2.2$). The nuclear modification factor, $R_{rm CuAu}$, of B mesons in Cu+Au collisions is consistent with binary scaling of measured yields in $p$+$p$ at both forward and backward rapidity.