This paper examines fresh spectral radius and norm limits for bounded linear operators which operate in Semi-Hilbertian spaces that emerge from fixed positive semi-definite operators which define the inner product structure. The extension of classical Hilbert space structures constitutes Semi-Hilbertian spaces which attract increasing research interest in modern functional analysis. The paper introduces basic concepts of operator theory in Semi-Hilbertian contexts focusing on their algebraic and geometric properties. The combination of improved inner product tools and operator inequalities leads us to derive precise upper and lower boundaries for the spectral radius of bounded linear operators. New norm inequalities show how operator norm and semi-inner products interact while displaying the connections to the metric given by the semi-definiteness of the operator. The proposed research findings transform into stability assessments for iterative methods and spectral examinations of differential operators. The derived bounds show superior performance compared to traditional limits on the Hilbert spaces through a comparative analysis thus delivering enhanced operational value.