“My landlord has requested I do not consume cannabis on the property where I rent my apartment, whether it be inside or outside, away from the building, etc. Is he legally allowed to prohibit smoking or consumption outside of the building? ”

With respect to the area outside of the building that is part of the property, or any common area within the building, then the landlord would indeed be allowed to prohibit smoking or vaping of any substance; this includes cannabis. If the outdoor area is beyond the property itself, then this would likely be considered a public space, and also a prohibited area to consume as per section 17 of the NB Cannabis Control Act.