Accurately estimating high-resolution carbon emissions is crucial for effective emission governance and mitigation planning. While conventional methods for precise carbon accounting are hindered by substantial data collection efforts, the rise of open data and advanced learning techniques offers a promising solution. Once an open data-based prediction model is developed and trained, it can easily infer emissions for new areas based on available open data. To address this, we incorporate two modalities of open data, satellite images and point-of-interest (POI) data, to predict high-resolution urban carbon emissions, with satellite images providing macroscopic and static and POI data offering fine-grained and relatively dynamic functionality information. However, estimating high-resolution carbon emissions presents two significant challenges: the intertwined and implicit effects of various functionalities on carbon emissions, and the complex spatial contiguity correlations that give rise to the agglomeration effect. Our model, OpenCarbon, features two major designs that target the challenges: a cross-modality information extraction and fusion module to extract complementary functionality information from two modules and model their interactions, and a neighborhood-informed aggregation module to capture the spatial contiguity correlations. Extensive experiments demonstrate our model's superiority, with a significant performance gain of 26.6\% on R2. Further generalizability tests and case studies also show OpenCarbon's capacity to capture the intrinsic relation between urban functionalities and carbon emissions, validating its potential to empower efficient carbon governance and targeted carbon mitigation planning. Codes and data are available: https://github.com/JinweiZzz/OpenCarbon.