Let $p$ be a prime and $m$ a positive integer. This paper investigates cyclic and self-dual codes of length $n$ over the local Frobenius non-chain ring $R=\mathrm{G}\mathrm{R}(p^3,m)[v]$, with $v^2=p^2\alpha$, $\alpha \in {\mathbb{F}}_{p^m}^{\ast }$, and $pv=0$. First, we characterize the algebraic structure of cyclic codes of arbitrary length over $R$. When $gcd(n,p)=1$, explicit generator polynomials are determined, and the corresponding dual and self-orthogonal structures are derived. A key result of this study is the proof that self-dual cyclic codes do not exist over $R$. In addition, the enumeration formula for cyclic LCD codes is given by $2^{e_1+\frac{e_2}{2}}.$ Several examples and tables are provided to illustrate the theoretical findings and the derived mass formulas for cyclic self-orthogonal and LCD codes.