(operationalizable) Cost measure for small problems
When we talk about complexity in computer science we are usually talking about asymptotic complexity. We are interested in characterizing the complexity of a process as the size of its inputs increase. This is useful because it abstracts away implementation-specific characteristics, such as processor speed or architecture or available memory. So while asymptotic analysis does, in this sense, concern itself with "large problems," its intent is not so much to treat large problems as it is to provide a meaningful metric for comparing algorithms in terms of time and space efficiency.

Kolmogorov complexity does not, to my knowledge, find applications in the practical analysis of programs or computational processes. It is an information-theoretic measure concerned with the minimal length of data as represented in a computational system. In so far as using it in any sort of "operational" analysis like those you've proposed, the big roadblock is that the Kolmogorov complexity of an arbitrary string is not, in general, computable!

It's difficult to pin down exactly what you're asking for, but it sounds more akin to counting CPU cycles required to execute a series of instructions. Back in the old days, when assembly language on punch-cards was the state-of-the-art, this was a fairly common technique used for optimization. 

As it applies to your example, adding two numbers would involve loading them, bitwise, into registers, performing the addition, and storing the result. If the addends required more bits to represent than the size of the registers, you'd have to do it piecewise, loading and storing intermediate results as needed. You can count the cycles needed to execute tasks of varying 'difficulties' (adding two 1-byte numbers vs. adding two 2-byte numbers, for example) and compare them to give some sense of how much more or less 'complicated' the execution of a given sequence of instructions is. 

It's not complexity in the traditional CS sense, but it would be a quantifiable, implementable metric. Donald Knuth has a program here that counts the cycles required to execute MMIX programs, as an example of such an implementation.