We held our first Canadian National Championship this year with much help from Eric Anderson and many others.
Promotion is just getting the word out, get playing in public parks, get people talking about it.
Sponsorship may be hard for a first tournament unless the sponsors are already familiar with Kubb. It may take a tournament or two to get them knowledgeable and ready to donate.
As for advice. Make sure you have enough sets for the matches as well as helpers to assist in setting up pitches. I'd recommend a quick-pitch system to make the time fly for setup. Make sure you have a way to let people know upcoming matches, etc. A bullhorn, PA system, whatever you have (a cone to talk through).
Have a backup plan in case of really bad weather. Will you hold it another day, play on in the rain/snow/hail/tornado what have you...(I don't recommend playing in a tornado or anything too nasty).
When the tournament has started, my wife gave me the best advice ever. Take a few moments while the matches are being played and just look. Enjoy what you have done. Especially your first tournament. It's a one time thing seeing it for the first time...cherish it.