Utility Theory/Marginal Rate of Substitution: Can the marginal rate of substitution be calculated for a point of the budget line? This a person's budget line with various points, and their consumption, C*, and their endowment e, which is worth $5000 (unimportant). Also shows is their initial indifference curve. The difference curve and budget line intersect at e and e'. 

Question: Why is the marginal rate of substitution of a, on the budget line, greater than C*?

I've been told any point that lies to the left of C* (the consumption optimum) has a marginal rate of substitution higher than the one at C* which is equal to the price ratio. Similarly, any point which lies to the right of C* has a marginal rate of substitution less than the one at C* which is equal to the price ratio.

The second indifference curve that C* would form part of is not shown in the diagram.

Why can I know the marginal rate of substitutions of points on the budget line if the marginal substitution is calculated on an indifference curve? Or, in this case, the MRS of C* is calculated by equating the slope of the indifference curve to the slope of the budget line. More to the point, how can I arrive at the answer that the MRS a > MRS C* if MRS can only be calculated on for a point on the indifference curve and not for a point on the budget line? Wouldn't all points on the budget have the same MRS implied by the line's slope?

There is something simple I'm not seeing.
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 $C^*$.