Through the tight-binding calculation, we demonstrate that magnetic and quantum confinements have a great influence on the low-energy band structures of one-dimensional (1D) armchair graphene ribbons. The magnetic field first changes 1D parabolic bands into the Hall-edge states which originate in the Landau wavefunctions deformed by one or two ribbon edges. The quantum confinement dominates the characteristics of the Hall-edge states only when the Landau wavefunctions touch two ribbon edges. Then, some of the Hall-edge states evolve as the Landau states when the field strength grows. The partial flat bands (Landau levels), related to the Landau states, appear. The magnetic field dramatically modifies the energy dispersions and it changes the size of the bandgap, shifts the band-edge states, destroys the degeneracy of the energy bands, induces the semiconductor–metal transition and generates the partial flat bands. The above-mentioned magneto-electronic properties are completely reflected in the low-frequency absorption spectra—the shift of peak position, the change of peak symmetry, the alteration of peak height, the generation of new peaks and the change of absorption edges. As a result, there are magnetic-field-dependent absorption frequencies. The findings show that the magnetic field could be used to modulate the electronic properties and the absorption spectra.