The proton mass can be calculated from lattice QCD to 1% accuracy; see Physical Nucleon Properties from Lattice QCD. A survey paper by Fodor and Hoelbling, Light Hadron Masses from Lattice QCD, discusses the state of the art of hadron mass prediction from lattice QCD. (also to 1% accuracy). For the meson spectrum, see Toward the excited isoscalar meson spectrum from lattice QCD. Extrapolation to physical parameter values, a lot of attention to detail, and a lot of computer time are needed to make this work.
Nonlattice techniques for mass prediction of mesons and hadrons include the Schwinger-Dyson equations; see, e.g., Masses of ground and excited-state hadrons.