Click-through rate (CTR) prediction plays as a core function module in various personalized online services. According to the data modality and input format, the models for CTR prediction can be mainly classified into two categories. The first one is the traditional CTR models that take as inputs the one-hot encoded ID features of tabular modality, which aims to capture the collaborative signals via feature interaction modeling. The second category takes as inputs the sentences of textual modality obtained by hard prompt templates, where pretrained language models (PLMs) are adopted to extract the semantic knowledge. These two lines of research generally focus on different characteristics of the same input data (i.e., textual and tabular modalities), forming a distinct complementary relationship with each other. Therefore, in this paper, we propose to conduct fine-grained feature-level Alignment between Language and CTR models (ALT) for CTR prediction. Apart from the common CLIP-like instance-level contrastive learning, we further design a novel joint reconstruction pretraining task for both masked language and tabular modeling. Specifically, the masked data of one modality (i.e., tokens or features) has to be recovered with the help of the other modality, which establishes the feature-level interaction and alignment via sufficient mutual information extraction between dual modalities. Moreover, we propose three different finetuning strategies with the option to train the aligned language and CTR models separately or jointly for downstream CTR prediction tasks, thus accommodating the varying efficacy and efficiency requirements for industrial applications. Extensive experiments on three real-world datasets demonstrate that ALT outperforms SOTA baselines, and is highly compatible for various language and CTR models.