Ischemic lesion segmentation and the time since stroke (TSS) onset classification from paired multi-modal MRI imaging of unwitnessed acute ischemic stroke (AIS) patients is crucial, which supports tissue plasminogen activator (tPA) thrombolysis decision-making. Deep learning methods demonstrate superiority in TSS classification. However, they often overfit task-irrelevant features due to insufficient paired labeled data, resulting in poor generalization. We observed that unpaired data are readily available and inherently carry task-relevant cues, but are less often considered and explored. Based on this, in this paper, we propose to fully excavate the potential of unpaired unlabeled data and use them to facilitate the downstream AIS analysis task. We first analyze the utility of features at the varied grain and propose a multi-grained contrastive learning (MGCL) framework to learn task-related prior representations from both coarse-grained and fine-grained levels. The former can learn global prior representations to enhance the location ability for the ischemic lesions and perceive the healthy surroundings, while the latter can learn local prior representations to enhance the perception ability for semantic relation between the ischemic lesion and other health regions. To better transfer and utilize the learned task-related representation, we designed a novel multi-task framework to simultaneously achieve ischemic lesion segmentation and TSS classification with limited labeled data. In addition, a multi-modal region-related feature fusion module is proposed to enable the feature correlation and synergy between multi-modal deep image features for more accurate TSS decision-making. Extensive experiments on the large-scale multi-center MRI dataset demonstrate the superiority of the proposed framework. Therefore, it is promising that it helps better stroke evaluation and treatment decision-making.