Automatic speech recognition (ASR) for low-resource languages such as Taiwanese Hokkien is difficult due to the scarcity of annotated data. However, direct fine-tuning on Han-character transcriptions often fails to capture detailed phonetic and tonal cues, while training only on romanization lacks lexical and syntactic coverage. In addition, prior studies have rarely explored staged strategies that integrate both annotation types. To address this gap, we present CLiFT-ASR, a cross-lingual fine-tuning framework that builds on Mandarin HuBERT models and progressively adapts them to Taiwanese Hokkien. The framework employs a two-stage process in which it first learns acoustic and tonal representations from phonetic Tai-lo annotations and then captures vocabulary and syntax from Han-character transcriptions. This progressive adaptation enables effective alignment between speech sounds and orthographic structures. Experiments on the TAT-MOE corpus demonstrate that CLiFT-ASR achieves a 24.88\% relative reduction in character error rate (CER) compared with strong baselines. The results indicate that CLiFT-ASR provides an effective and parameter-efficient solution for Taiwanese Hokkien ASR and that it has potential to benefit other low-resource language scenarios.