I know some companies in Hawaii and grand cayman give you 45 min submarine tours for $400
Is there a way to get a more authentic experience, like being on a sub for an entire weekend?
Or do you need to have a brother that is an ex-navy seal?
The Navy does have programs that will allow journalists, community leaders, and other individuals to spend a great deal more time on a submarine than just a few minutes. However, these are not exactly common, and not for a week. For instance, in this story, educators from Chicago were able to spend a day on a fast attack sub.
The USAF also offers flights in different aircraft to civilians under the same goodwill program.
I am not personally aware of any programs in foreign services, however it is very likely that many of the more modern all volunteer forces have similar programs to foster goodwill and a favorable public image.
What sort of submarine are you looking for? travelscience.com lists 2 tour companies that offer deep-sea submarine expeditions to the titanic, bismark or black smokers. The advenbtures.ca tour is 12 days long and will set you back ~$60.000, but im not sure how much fun it would be in such a small can:
AFAIK it takes ~2h to dive to the titanic, and 2h to get back up.