UN Headquarters Visitor Center where the stamps and post office is located starting June 2013. A ticket costs 16 US Dollars; however the tour tickets are NOT sold on the spot.
Another notice from the United Nations Postal Administration says: "As part of the renovation project for the General Assembly Building at United Nations Headquarters in New York, the UNPA Stamp Shop has been relocated to a temporary Visitor's Centre in the Dag Hammarskjöld Library on the south end of the campus. Due to security limitations, visitors are not allowed into the Centre without prior notice of arrival. To schedule a visit to the UNPA Stamp Shop, please call 212-963-7698 or 1-800-234-8672 for access instructions. The updated UNPA Stamp Shop will reopen to unannounced visitors in the Fall of 2014." Send me an email if you have successfully scheduled a visit without paying the guided tour. I will update the information here.
You can still forward your prepared mails to the United Nations Headquarters to be released in the mail streams there by following the steps mentioned in a previous post.