If the previous tenants to our apartment broke the microwave but the landlord refuses to fix it for us, what are our rights?

Check the lease to be sure what it is that the LL was required to deliver to you in working order. Also see the link below to get a better understanding as to what a residential LL is required to do.

Dear New York Tenant:
If the lease discussed working appliances, then likely the landlord should replace the microwave. Otherwise, you may have a problem since a defective microwave oven will not carry any significant weight in a legal dispute with the landlord if the stove works.
Your lease may have described items for the landlord to repair after you took possession, so review your lease.

Do you have a written lease? If so, does it state that the apartment comes with a microwave and who is responsible for maintenance of the appliances? If the answer to any of these questions is no then you do not have a claim to a working microwave. If you are a month to month tenant then you rented the apartment as it was. Your failure to test the microwave prior to taking possession does not mean the landlord is required to fix it. If you have a written lease the lease should address appliances and who is responsible for maintaining them Review the terms of your lease to determine if a microwave was included and if the landlord is responsible for it.