There are actually three passes that you could take between Hancock and Alpine Station. I have not gone over any of them but have been on both sides of the divide and talked to some Jeepers.
The shortest way is for hiking (mountain climbing?) only. This is often called Alpine Pass but is really Altmont Pass, I believe. A lot of 4 Wheelers take Williams Pass to the south. And I think Hancock Pass (the southernmost) is a bit easier. I wouldn't do either one in anything except a Wrangler or equivalent. And as was said, only in mid-July to mid-September.
See if this map link works: