Catalyzes the reversible oxidation-reduction of the R-enantiomer of free methionine sulfoxide to methionine. Specific for free L-methionine-(R)-S-oxide. [thioredoxin]-disulfide + H2O + L-methionine = [thioredoxin]-dithiol + L-methionine (R)-S-oxide Belongs to the free Met sulfoxide reductase family. Extended N-terminus.