Curvilinear structure segmentation (CSS) plays a vital role in industrial applications, including medical imaging and structural health monitoring. Recently, the strong capacity of the Segment Anything Model (SAM) has inspired its downstream application in CSS tasks. To adapt SAM to CSS tasks, previous methods heavily rely on a certain number of samples and costly pixel-level annotation, which are hard to access for a new scenario. Considering this, the goal of our work is to adapt SAM in a very cost-effective setting where only a single unlabeled image is given. This is far more challenging than the typical supervised, unsupervised, or self-supervised learning manner that needs a large number of training samples. To tackle this problem, we propose a finetuning-free SAM for curvilinear structure segmentation, called \textbf{c}urvilinear-\textbf{a}ware \textbf{pro}mpt learning (\emph{CaPro}), which aims to automatically learn visual prompts via a single unlabeled image. In the first stage, we generate extensive curvilinear structures and oriented sub-curvilinear box annotations. To increase the realism of generated curvilinear structures, we adapt these structures into real image domains via the Fourier Transform using a single real-world unlabeled image. Now, these adapted images can be used to train our oriented sub-curvilinear detector. In the second stage, we propose the curvilinear-aware discrete representation matching to filter those unreliable detection results. Afterward, these reliable detection results can be converted into informative prompts, contributing to the cost-effective SAM adaptation to CSS tasks. Experiments demonstrate the effectiveness of \emph{CaPro} on medical image and crack segmentation tasks. Code and dataset will be publicly available.