The procedure is actually quite easy and quite cheap. I've done it very recently using Lee dies and case length gauges (both very functional and inexpensive).
First, get your 7.7 jap size die set up in the press. Now, run your .30-06 case through the size die until the going gets tough (that is, the case shoulder makes contact with the die). At this point, you've removed the spent primer and "necked up" from 308 to 312. Now that the neck is expanded, Lee's case length gauge/trimmer will fit in the case. I trim the case way down, perhaps 1/4" from the shoulder. Now put the case in the press again, this time forcing it all the way into the die. Withdraw the case and the case neck will be way too long, trim again and you're done.
I just formed some 7.7 jap case yesterday, but have never fired formed 7.7 cases before. Take out your calipers and you'll note that the base of the .30-06 is indeed smaller than the 7.7. I'm told that when you fire the formed 7.7 cases the first time, you'll get uncontrolled expansion at the base of the case that makes for some ugly buldging. I've heard it recommended that you wrap some scotch tape around the base of the case before the first firing to "take up the slack" in the chamber; supposedly this contributes to more uniform expansion and does away with the buldgy look.
Finally, a formed .30-06 case is by no means a perfect 7.7 case, and I've heard that some people only used the form cases for light cast bullet loads. Then again, Arisakas are known for their strength and I've heard of many people who use formed cases with standard loads with no ill effects.
I'm also told that formed cases won't last as long as true 7.7 cases, but given that 7.7 Jap cases cost a fortune and once-fired 30-06 cases are free pick-ups, I don't find this too troubling.