Specifically hydrolyzes 7-methylguanosine monophosphate (m(7)GMP) to 7-methylguanosine and inorganic phosphate. The specific activity for m(7)GMP may protect cells against undesired salvage of m(7)GMP and its incorporation into nucleic acids. Also has weak activity for CMP. UMP and purine nucleotides are poor substrates (By similarity). H2O + N(7)-methyl-GMP = N(7)-methylguanosine + phosphate CMP + H2O = cytidine + phosphate a ribonucleoside 5'-phosphate + H2O = a ribonucleoside + phosphate Monomer. Belongs to the pyrimidine 5'-nucleotidase family.