You don’t have to show the apartment if you don’t want to. The landlord must give you notice as to when he wants to show the apartment. You can make a counter offer to show it on Sunday. Even if this is verbal follow up with a certified letter saying as per our conversation … this way you have written proof that you were trying to give him access to the apartment on your terms. As a general rule the landlord will not sue you as it will cost thousands of dollars in legal fees and as you are a renter chances are he will have trouble collecting the judgement. All he may do is hold your security, but then you can sue him in small claim court. You don’t need a lawyer to fight your case. The landlord may decide to return the security especially if he has to hire a lawyer, or doesn’t want to waste his time in court.
This is all assuming that you are pay your rent on time. In NYC this would be called harassment and you could report him to the city,