Motivated by recent advancements in the Josephson diode effect and unconventional $p$-wave magnets (UPMs), we investigate the supercurrent in UPM/superconductor Josephson junctions by the Green's function method. The simultaneous presence of an out-of-plane Zeeman field and Rashba spin-orbit coupling within the UPM breaks the conventional current-phase relation $I\left(\phi \right)=-I(-\phi )$, yielding a Josephson diode effect with high rectification efficiency. The diode polarity and efficiency exhibit systematic oscillations as functions of the Néel vector orientation and strength, as well as Rashba spin-orbit coupling. We identify the conditions necessary for the emergence of the Josephson diode effect through symmetry analysis. Furthermore, the current-phase relation and the Josephson diode effect are sensitive to the orientation of the Néel vector, offering an alternative scheme for its electrical detection. These findings enhance the understanding of the Josephson diode effect in UPM-based Josephson devices and provide a novel detection method for the Néel vector in UPMs.

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• Josephson effect
• Spin-orbit coupling
• Altermagnets
• Josephson junctions
• Bogoliubov-de Gennes equations
• Green's function methods