How Create a list of the first 10,000 digits of Pi and sum it?

Does this do what you want:

Total @ RealDigits[Pi, 10, 10^4][[1,259;;7233]]

You can use the four-argument of form of RealDigits:

RealDigits[$x, b, len, n$] gives len digits starting with the coefficient of $b^n$.

m = 259;
n = 7233;
Total@First@RealDigits[Pi, 10, n - (m - 1), -(m - 1)]

31574

RealDigits[Pi, 10, 10000] // First // Take[#, {259, 7233}] & // Total

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