Let us define a function
$$f(k) = (-1)^{k+1}\left(1 - \sum_{j=2}^k (-1)^j \zeta(j)\right)$$
for positive integers $k\ge 2$ and, $f(1)=1$ by definition, and $\zeta$ is the Riemann Zeta function.
Is there a known method to extend this function to real numbers $r>1$ (or even $r>0$?).
Not in the way you mean: in order to extend a function you would want to have it defined on a set with a limit point, but your definition is only on a discrete subset of $\Bbb R$, i.e. a set of natural numbers. Even if you found an extension function, there is no guarantee of uniqueness, since the identity theorem requires a set with a limit point.
If you mean just any choice, yeah, that's easy, it's in textbooks: See Theorem 15.13 in Rudin: Real and Complex Analysis for example.