Breast cancer (BRCA) has an undesirable prognosis and is the second most common cancer among women after lung cancer. A novel mechanism of programmed cell death called cuproptosis is linked to the development and spread of tumor cells. However, the function of cuproptosis in BRCA remains unknown. To this date, no studies have used machine learning methods to screen for characteristic genes to explore the role of cuproptosis-related genes (CRGs) in breast cancer. Therefore, 14 cuproptosis-related characteristic genes (CRCGs) were discovered by the feature selection of 39 differentially expressed CRGs using the three machine learning methods LASSO, SVM-RFE, and random forest. Through the PPI network and immune infiltration analysis, we found that PRNP was the key CRCG. The miRTarBase, TargetScan, and miRDB databases were then used to identify hsa-miR-192-5p and hsa-miR-215-5p as the upstream miRNA of PRNP, and the upstream lncRNA, CARMN, was identified by the StarBase database. Thus, the mRNA PRNP/miRNA hsa-miR-192-5p and hsa-miR-215-5p/lncRNA CARMN ceRNA network was constructed. This ceRNA network, which has not been studied before, is extremely innovative. Furthermore, four cuproptosis-related lncRNAs (CRLs) were screened in TCGA-BRCA by univariate Cox, LASSO, and multivariate Cox regression analysis. The risk model was constructed by using these four CRLs, and the risk score = C9orf163 * (1.8365) + PHC2-AS1 * (−2.2985) + AC087741.1 * (−0.9504) + AL109824.1 * (0.6016). The ROC curve and C-index demonstrated the superior predictive capacity of the risk model, and the ROC curve demonstrated that the AUC of 1-, 3-, and 5-year OS in all samples was 0.721, 0.695, and 0.633, respectively. Finally, 50 prospective sensitive medicines were screened with the pRRophetic R package, among which 17-AAG may be a therapeutic agent for high-risk patients, while the other 49 medicines may be suitable for the treatment of low-risk patients. In conclusion, our study constructs a new ceRNA network and a novel risk model, which offer a theoretical foundation for the treatment of BRCA and will aid in improving the prognosis of BRCA.