If you feel adventurous, it's probably a pretty simple fix. Id say one of the discs has jumped off or mis-aligned on it's mechanism. In short, you would need to remove the movement, remove the hands, remove the dial. By this point the discs are in plain view and if you are mechanically minded, the problem will probably pretty obvious. Also spend some time studying the Seiko tech manuals for the 4006 before doing the job, they make it pretty clear how the day-date system works.
Words of advice, invest in quality plastic tweezers and remember to cover the dial when removing hands, its pretty easy to trash a dial with tools. If you think you will do this kind of stuff more often, invest in a good set of screwdrivers and a "presto" hand puller for your first bits of kit.