Compilers, while essential, are notoriously complex systems that demand prohibitively expensive human expertise to develop and maintain. The recent advancements in Large Language Models (LLMs) offer a compelling new paradigm: Neural Compilation, which could potentially simplify compiler development for new architectures and facilitate the discovery of innovative optimization techniques. However, several critical obstacles impede its practical adoption. Firstly, a significant lack of dedicated benchmarks and robust evaluation methodologies hinders objective assessment and tracking of progress in the field. Secondly, systematically enhancing the reliability and performance of LLM-generated assembly remains a critical challenge. Addressing these challenges, this paper introduces NeuComBack, a novel benchmark dataset specifically designed for IR-to-assembly compilation. Leveraging this dataset, we first define a foundational Neural Compilation workflow and conduct a comprehensive evaluation of the capabilities of recent frontier LLMs on Neural Compilation, establishing new performance baselines. We further propose a self-evolving prompt optimization method that enables LLMs to iteratively evolve their internal prompt strategies by extracting insights from prior self-debugging traces, thereby enhancing their neural compilation capabilities. Experiments demonstrate that our method significantly improves both the functional correctness and the performance of LLM-generated assembly code. Compared to baseline prompts, the functional correctness rates improved from 44% to 64% on x86_64 and from 36% to 58% on aarch64, respectively. More significantly, among the 16 correctly generated x86_64 programs using our method, 14 (87.5%) surpassed clang-O3 performance. These consistent improvements across diverse architectures (x86_64 and aarch64) and program distributions (NeuComBack L1 and L2) validate our method's superiority over conventional approaches and its potential for broader adoption in low-level neural compilation.