I am planning to drive over the Curonian Spit from the Lithuanian side in 2013. I would love to travel on into the Russian side and visit Kaliningrad. Is there a border crossing on the Spit? If not, will I have to travel back up the Lithuanian side and find an alternative crossing into Russian/Kaliningrad?
If you are not a Russian citizen, you need a Russian visa. This has to be arranged before going there. They won't issue a visa at the Nida border.
As you say you're going to pass the border at Curonian Spit, you'll be in Klaipeda before this. @Marcel C. already said about the Nida station, but I want to add some information which can be useful for you and others who will view this question.
So, you have two different ways to get from Klaipeda to Curonian Spit and Kaliningrad with your car:
I should note that where are other way to get to the Kaliningrad from Klaipeda, if you don't want to use your car - international buses are operating between two cities on daily basis.
You can find information here (site about the Kaliningrad bus station) and here (site about the international bus line from Kaliningrad). Note that various buses have various stops - some of them will move you across the Klaipeda's bay, some of them will not.