State-of-the-art object detection and segmentation methods for microscopy images rely on supervised machine learning, which requires laborious manual annotation of training data. Here we present a self-supervised method based on time arrow prediction pre-training that learns dense image representations from raw, unlabeled live-cell microscopy videos. Our method builds upon the task of predicting the correct order of time-flipped image regions via a single-image feature extractor followed by a time arrow prediction head that operates on the fused features. We show that the resulting dense representations capture inherently time-asymmetric biological processes such as cell divisions on a pixel-level. We furthermore demonstrate the utility of these representations on several live-cell microscopy datasets for detection and segmentation of dividing cells, as well as for cell state classification. Our method outperforms supervised methods, particularly when only limited ground truth annotations are available as is commonly the case in practice. We provide code at https://github.com/weigertlab/tarrow.