As far as I am aware, there is no evidence existing that supports the Germans expected a Japanese attack on Russia.
However, I believe that Hitler knew at the time he made his speech declaring war on the U.S. on December 11, 1941, that Germany was in real trouble at the gates of Moscow. For this reason it is my opinion that Hitler declared war on the U.S. hoping for help from the Japanese.
The following supports this opinion:
Many historians and writings have stated Hitler did this to show loyalty to his Japanese allies, simply making official the fighting that had been going on between the U.S. and Germany in the Atlantic since the fall of France or as some sort of retalliation for Lend Lease to German enemies.
Short of saying it in a speech, Germany couldn't DO anything more than they were already doing against the U.S. What is the point in giving America justification to join England in a bombing campain if it doesn't help the German situation at the time? The answer is there is no point. Making dumb moves like that fits the Hitler of the comming desperation times but not of late 1941. The Hitler of 1941 still worked situations to his benefit in most cases.
Hitler knew going into Operation Barbarosa, Germany could not win a war of attrition with the Russians. He has been quoted saying that he expected complete victory before the end of Summer 1941. When it became obvious this was not possible, he was well aware that Germany would not win a prolonged war. However, he still thought that victory for the most part, could be secured during Barbarosa as late as early December, 1941.
It was during the very week of the Japanese attack on Pearl Harbor, operation Barbarosa ground to a halt at the gates of Moscow. Right after the stall (December 5th and 6th, 1941), the Soviets attacked German forces around Moscow with forces that included 70 fresh divisions of well equipped, Siberian troops. Troops that the German High Command didn't think were available.
One day before Pearl Harbor, the temperature along the Moscow front dropped to -38f. Lubricants for German guns and equipment had been jelling up for the past couple of nights due to temperatures around -25f all along the front, affecting combat worthiness drastically. From the 5th on, the Soviets were making small gains (a mile or less) in several areas. On the 6th, Soviet Thitieth Army broke into Third Panzer Group's left flank northeast of Klin to a distance of eight miles, almost creating a complete breakthrough.
German generals were reporting all along the line about the fresh, well trained and equipped Russian Siberian forces. For the next several days German forces were retreating west from Moscow. Early on, it looked at times like the routs the German were accustomed to creating but they were the ones on the run. On the 10th of December, Guderian characterized his Second Panzer Army as a scattered assemblage of armed baggage trains slowly wending their way to the rear.
Hitler knew everything that was happening along the front and was updated several times a day. He knew his armies were stalled and falling back. He also knew that unless something changed quickly, the objectives of Barbarosa would not be accomplished in 1941. In fact, some German generals later stated that as of December 9th and 10th both Hitler and they realized Germany may well loose the war eventually unless they could somehow turn the situation quickly.
In addition, Hitler was aware that his nation's strategic fuel reserves were low and some of the armies in Russia were well short of normal strength. He had plans for how to get troop strengths back up but the oil situation alone threatened everything.
Even with the armies back to full strength, the German oil reserves were too low and replenshment too slow to be able to adequately replace supplies by Spring 1942 for another all-out offensive like Barbarosa.
In order to turn the situation immediately, Hitler was looking for ways to change things on the Russian front. He hoped declaring war on the U.S. would induce Japan to reciprocate the gesture by declaring war on the Soviets. At the least he hoped to persuade his ally to create concerns along the Russian boarder.
New concerns to the East may cause Stalin to re-think his decision to pull Siberian divisions from that front to reinforce Moscow. Hopefully, he would then send back at least part of the units he had taken from that area for Moscow defense since August, 1941.
The Soviets and Japanese were not allys by any stretch. In fact, Stalin had feared a Japanese attack on the Russian Eastern border since the war bagan. It was very reluctantly--and possibly out of desperation--that he had moved forces from there to the Moscow area to begin with.
If successful, Hitler likely would have thought his forces could once again go on the offensive. He was still under the mistaken impression the Russians were completely out of reserves and the original move by Stalin, though successful, would likely not have put his forces on the defensive without the additional bad fortune of the onset of really cold weather. Nazi climate research had indicated the the really cold weather of a Russian Winter was still a couple of weeks away when it hit.
Other than his hopes for immediate help with the Soviet situation--which would have been admitted to NO ONE by the Nazi regime--it is unexplainable that Hitler would just up and declare war on America with no real upside for his country at the time.
Hitler rolled the dice in hopes it would help the cause. But the Japanese didn't bite!