Utility Theory/Marginal Rate of Substitution: Can the marginal rate of substitution be calculated for a point of the budget line?
Your claim that "MRS can only be calculated on for a point on the indifference curve and not for a point on the budget line" reveals a misunderstanding. Note that every point lies on some indifference curve. So each point on the budget line also lies on some indifference curve. We also know that all indifference curves are convex, since the utility function has Cobb-Douglas form.

At $C^*$, the optimal bundle, the indifference curve has to be tangent to the budget line, so it has the same slope as the budget line. At point $a$ the indifference curve (by convexity) intersects the budget line from above and therefore must be steeper than the budget line. Thus the MRS at $a$ is larger than the MRS at $C^*$.