From Matsue east to Kamakura....

We plan to travel from Matsue to Kamakura and need to find a route that's economical and includes the worthwhile sites. Would it be best to visit Adachi Art Museum etc.along the way instead of a day trip from Matsue?
If we hire a car in Matsue to tour the sights is it economical to leave it in say,Tottori and take a train onwards then towards Kamakura?
Is there a train pass we can use along the route to Kamakura? We don't need to go through Tokyo - we'll be going there after Kamakura anyway.
Hope someone can help.

How much time do you have. If you have just one day, I recommend to use the limited express train and shinkansen via Okayama. If you have several days to spend along the way, you could travel along the Sanin coast with stops in Kinosaki Onsen and Amanohashidate.

Would it be best to visit Adachi Art Museum etc.along the way instead of a day trip from Matsue?

Only if you travel via Tottori, it makes sense to visit the museum along the way. But not too much time is saved by doing so. If you have lots of luggage, just doing a side trip might be more comfortable.

If we hire a car in Matsue to tour the sights is it economical to leave it in say,Tottori and take a train onwards then towards Kamakura?

It depends on how many people you are and how you would be traveling otherwise. But at least, it would be an attractive way of travel. Access to the Adachi Art Museum would be convenient. You could also consider dropping the car off even later along the way, depending on your route.

Is there a train pass we can use along the route to Kamakura?

The Japan Rail Pass.