Japanese Multiplication simulation - is a program actually capable of improving calculation speed? Or am I doomed from the start?
When considering algorithms for multiplying large numbers, the first think to keep in mind is the asymptotic complexity. Generally speaking, algorithms with better (smaller) asymptotic complexity are faster, though this only goes so far. So the first thing to do to find out whether Japanese multiplication is "worthwhile" is to calculate its asymptotic complexity. You have to be careful with respect to the model of computation, but in your case it probably won't make a big difference; probably your algorithm has "bad" asymptotic complexity.

The second thing to notice is that the multiplication logic is already there in the CPU, and no code that you can write will be faster than calling the built-in multiplication instruction. Of course, this is only good for multiplying machine words (or multiwords, in some cases). That means that your algorithm had better take advantage of the multiplication capabilities of your processor. While not affecting the asymptotic complexity, it does considerably affect the constants. All fast multiplication algorithms use machine multiplication instructions.

The best way to know whether the algorithm is any good in practice is to program and optimize it. Use a language like C which produces fast code, and then you can compare it to libraries such as GMP. This is the only convincing way to claim that your algorithm is better than existing ones, though you should expend some effort on optimization.

Finally, you claim that Japanese multiplication is better than CPU multiplication algorithms since it only uses addition. Textbook multiplication algorithm also only use addition, since everything is in binary. But probably this can be optimized further at the expense of using more gates, by having logic for small multiplications. You can look up what algorithms CPUs use, perhaps they're similar to Japanese multiplication.