Automated essay scoring (AES) aims to score essays written for a given prompt, which defines the writing topic. Most existing AES systems assume to grade essays of the same prompt as used in training and assign only a holistic score. However, such settings conflict with real-education situations; pre-graded essays for a particular prompt are lacking, and detailed trait scores of sub-rubrics are required. Thus, predicting various trait scores of unseen-prompt essays (called cross-prompt essay trait scoring) is a remaining challenge of AES. In this paper, we propose a robust model: prompt- and trait relation-aware cross-prompt essay trait scorer. We encode prompt-aware essay representation by essay-prompt attention and utilizing the topic-coherence feature extracted by the topic-modeling mechanism without access to labeled data; therefore, our model considers the prompt adherence of an essay, even in a cross-prompt setting. To facilitate multi-trait scoring, we design trait-similarity loss that encapsulates the correlations of traits. Experiments prove the efficacy of our model, showing state-of-the-art results for all prompts and traits. Significant improvements in low-resource-prompt and inferior traits further indicate our model's strength.