Dear can I withhold rent?
You could withhold rent, if you have the belief that somehow your not paying rent at this time, will manipulate your landlord into starting the legal process against the neighbor, but if you had an attorney, I believe that your attorney would suggest that is not a wise course of conduct, as the upshot is that you would be in court defending your right to keep your home, before the landlord could get the lease violation holdover going against the neighbor.
You have made significant forward progress as your landlord has yielded to your desire that the neighbor is informed of the need to carpet the apartment. So you should let the process evolve. This may be a good time to secure the services of an attorney to nudge the landlord and its attorney along. You will necessarily become involved in the lawsuit the landlord starts against the neighbor, as you are a witness to the events of the intrusion of the noise into your apartment caused by the upstairs tenant's breach of an obligation of tenancy.
Your attorney would suggest that you continue to monitor the noise, inform the landlord and or management, in writing, by letters sent from the post office, with mailing receipts for certified mail and by regular mail (making certain that you keep duplicate originals of all your letters and keep safe the mailing receipts) of the continuing events, and request that you are updated as to the progress the landlord's lawyer is making to bringing their case to court.
If you are shut out from updates and or the landlord stalls, then your attorney will guide you on a course of conduct that may secure your tenancy rights without risking that you become a new landlord target.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.
Dear "Can I Withhold Rent,"
I agree with much of what Mr. Smollens said in that, in this situation, where the landlord has already agreed to write to the noise making tenant, it would be preferable to hire an attorney to write a demand letter to the landlord, advising that if it doesn't take legal action immediately against your neighbor, you'll have no choice but to withhold your rent.
You should be aware that if you decide to withhold your rent (usually tenants will withhold the entire month's rent for issues like this) then you will likely be sued in court for non-payment of rent, which will likely cause you to become blacklisted from renting an apartment in the future. Once a tenant is named a respondent in a proceeding in housing court and the case is calendared, their name becomes part of the housing court's records, which are sold to tenant screening bureaus. Accordingly, if you can avoid going to court you should--especially if you intend to rent elsewhere in the future.
If the noise persists, prior to going to housing court, you should consult an attorney who may be able to remedy this issue while keeping you off the blacklist. Good luck!