The 1911 census states that WJGrant arrived 1905 - this appears to be the case. He arrived at St. John, New Brunswick on 11/3/1905 aged 22 (i.e. bc.1883) occupation Traveller. He is shown as a Commercial Traveller in 1911 which fits.
The census also shows that he was born in the month of July (although, as you say, year given as 1881).
The William Joseph Grant, son of Joseph & Harriett Mary (nee Smith) was indeed born in July:
Whilst birth registered as William Joseph Grant in Sept.qtr.1883 his baptism was as follows:
William Joseph Cullingworth Grant, born 27 July 1883, baptised 9 September 1883 St. Saviour, Southwark, son of Joseph and Harriett Mary. Abode East Dulwich, and Joseph was a Commercial Clerk.
So, from his arrival in Canada aged 22, and birth details, it does indeed appear that your man was born in July (27th) 1883 (not 1881), son of Joseph Grant and Harriett Mary (nee Smith).
Annette