Simultaneous transmitting and reflecting (STAR) reconfigurable intelligent surface (RIS) technique has recently received considerable attention due to its omni-directional radiation capability. In this paper, motivated by the interference-mitigation-based (IMB) and signal-enhancement-based (SEB) designs, we introduce an innovative STAR-RIS assisted simultaneous-signal-enhancement-and-interference-mitigation (SSEIM) design in non-orthogonal multiple access (NOMA) multiple-input single-output cellular communication networks. Our objective is to maximize the system spectral efficiency (SE) by jointly optimizing the reflection and transmission phase shifts at the STAR-RIS, the precoding matrix of BSs, and the power allocation factors of NOMA users. We propose a low-complexity simultaneous enhancement and mitigation algorithm. Furthermore, by exploiting the manifold optimization technique, we introduce the Riemannian conjugate gradient algorithm to solve the non-convex subproblems with unit modulus constraint. Our analysis reveals that the proposed SSEIM design exceeds the traditional RIS-aided SEB and IMB designs.