Alright. I'll explain it now anyway. I couldn't be bothered before

In the greatest of kingdoms, - Hilda's a Lindblum lass.
With work, I reside, - She lives with Cid, who is always working.
To help out the royalty,
With their work and their pride. - - Either helping literally in his work, or just being there when he comes home. Also Cid could say "I have a wife" and be full of pride.

I'm not on the outside,
As my work is within, - Being with the Regent, she would probably stay mostly inside the Grand Castle.
My name is the top,
Of the others aboard. Finally, you have the Hilda Garde airship. "Hilda" would most likely be written on it, above the crewmen.