The old Priory (next to St Marys, the parish church) used to be a secondary school for boys (maybe girls as well ?) in the 1940, 50's and I think into the 1960's ... but that's not Monk Street, its Priory Street. Monk Street runs from Whitecross St, past the end of the churchyard, and across the junction of Priory St and Dixton Rd, and uphill towards the girls school, where it changes into Hereford Road !
(If you Google map Monmouth, the names of the roads on it are WRONG ... I know; I live there !)
I'll be in town tomorrow and will ask an archaeologist friend who used to go to the Priory School if he knows of any school in Monk Street in the 1880's ... but I dont think there was one ... he's also got copies of all of Keith Kissacks excellent books about Monmouth which might give the answer (mine are all packed as I'm moving house !)