Does the UN have an official anthem or hymn?

No, the UN does not have an official anthem or hymn. In 2000, the Department of Public Information issued fact sheet no. 9 which states the following:

Though many songs have been written about the United Nations or on related themes, there is no official anthem or hymn for the Organization. One such song, or hymn, was written and performed at the United Nations, on 24 October 1971, on the occasion of the 25th anniversary of the Organization, by maestro Pablo Casals of Spain. The words were written by poet W.H. Auden of the United Kingdom.

The words of the song called, A Hymn to the UN, is also contained in the fact sheet. Two different scores for the song - one for voice and piano and one for a wind band - are attached below.