The PHENIX experiment at the Relativistic Heavy Ion Collider (RHIC) has measured ϕ meson production and nuclear modification in asymmetric Cu+Au heavy-ion collisions at 200 GeV at both forward Cu-going direction (1.2 < y < 2.2) and backward Au-going direction (-2.2 < y < -1.2) rapidities. Due to its very short lifetime, the ϕ meson is an excellent probe for studying the hot and dense state of nuclear matter, referred to as the quark-gluon plasma (QGP), that is produced in high-energy heavy-ion collisions, such as those at RHIC. Furthermore, the absence of strong interactions between muons and the surrounding hot hadronic matter makes the ϕ decay channel particularly useful for studying nuclear matter effects on ϕ meson production. Additionally, the rapidity dependence of ϕ meson production in asymmetric heavy-ion collisions provides a unique means of accessing the entanglement of hot and cold nuclear matter effects. However, the large combinatorial background produced at forward and backward rapidities in heavy-ion collisions results in a very challenging environment for extracting the ϕ meson signal. Accordingly, previous measurements at RHIC were limited to smaller collision species, p+p and d+Au. In this paper, a procedure for modeling and removing the backgrounds is detailed, and the first ϕ meson measurement at forward and backward rapidites in heavy-ion collisions at RHIC is presented. The ϕ meson invariant yield and nuclear-modification factor are reported as a function of the number of participating nucleons, rapidity, and transverse momentum in the kinematic region 1.2 < |y| < 2.2 and 1 < p_T < 5 GeV/c. Results of this analysis provide insight into the mixture of hot and cold nuclear matter effects on ϕ meson production in asymmetric heavy-ion collisions, bringing scientists one step closer to understanding the QGP.