Sorry, tenant or not, you have no rights. If the property owner posts a state-legal No Weapons sign, that applies to ANYONE that enters the building. 25 years in the real estate business as a broker, educator, landlord and consultant is my background on this issue.
No, I'm not a lawyer (thank goodness), but tenants (residential or commercial) don't have the same rights as property owners.
Depending on the local market, he may ignore it, or it may give him the opportunity to give you the boot. And from a legal liability standpoint, he is unlikely to give you a pass. If there is an issue with someone else with regards to a firearm, he puts himself in legal jeopardy by giving you permission to carry.