The H is silent in "honour". You use "an".

I think that this is one of those rules that has changed over the years, but the more common use nowadays (at least in my part of the world) would be "an honour". E.g. It is an honour to meet you. It would be an honour to be accepted into this school.

This is a tricky one, but I personally say "an honor" and teach my students to do the same.

it is a very much debated topic, we used to say A honour because of H not being a vowel. there would be elocution lessons on how to say A honour (H following A) it is a little bit of a tongue twister many people cannot say A honour because switching from A to H produces the 'AN' sound. now we say an honour because of expiring air when saying the H after A.