Since such a post at least implies, and possibly explicitly says, that the person is acting poorly or improperly, perhaps even illegally, as a tenant, it would be potentially defamatory. A suit for defamation would be possible. But that would require the person "named" to establish that the comments were statements of fact rather than opinions, and that they were false. It would also require proof of damage to reputation, or else proof that the statements were in one of the categories that local law treats as defamation per se Those categories vary by jurisdiction, but a statement that a person i guilty of a crime usually qualifies.
In the US at least, there is no general right of privacy. There are various specific rights which can form the basis of civil suits, but these vary quite a bit from state to state. I am not sure just what the law on such matters is outside the US. Such a situation might fit a false light claim, but in at least some states false light is taken as just another term for defamation, or defamation by implication, and requires the same proof. The situation described in the question would not fit a claim of private facts nor one of infringement of personality rights. If the picture were taken by sneaking up and shooting through the tenant's window it might fit a claim of intrusion on seclusion but not if it were a previously published picture, or one taken in a public place.
In short, what recourse the person "named and shamed" would have depends on the exact facts, and on the jurisdiction where this occurs.