Due to the rapid increase in video traffic and relatively limited delivery infrastructure, end users often experience dynamically varying quality over time when viewing streaming videos. The user quality-of-experience (QoE) must be continuously monitored to deliver an optimized service. However, modern approaches for continuous-time video QoE estimation require densely annotating the continuous-time QoE labels, which is labor-intensive and time-consuming. To cope with such limitations, we propose a novel weakly-supervised domain adaptation approach for continuous-time QoE evaluation, by making use of a small amount of continuously labeled data in the source domain and abundant weakly-labeled data (only containing the retrospective QoE labels) in the target domain. Specifically, given a pair of videos from source and target domains, effective spatiotemporal segment-level feature representation is first learned by a combination of 2D and 3D convolutional networks. Then, a multi-task prediction framework is developed to simultaneously achieve continuous-time and retrospective QoE predictions, where a quality attentive adaptation approach is investigated to effectively alleviate the domain discrepancy without hampering the prediction performance. This approach is enabled by explicitly attending to the video-level discrimination and segment-level transferability in terms of the domain discrepancy. Experiments on benchmark databases demonstrate that the proposed method significantly improves the prediction performance under the cross-domain setting.