Altermagnets provide a new route to engineer superconducting circuits without magnetic fields. We theoretically study the Andreev bound state (ABS) spectrum of a finite-width altrmagnet-based Josephson junction and show how the $d$-wave altermagnetic symmetry and geometric confinement shape its low-energy excitations. We find a clear distinction between the two $d$-wave symmetries: $d_{x^2-y^2}$ order produces spin splitting, whereas $d_{xy}$ order preserves spin degeneracy and exhibits splitting of the ABS spectrum induced by intermode hybridization. Leveraging these novel features, we propose applying a transverse electric field to tune the system and realize a magnetic-field-free, parity-protected superconducting qubit that we call altermon.