Recently, constructions of linear complementary pairs (LCPs) of codes and linear complementary dual (LCD) codes on function fields have attracted considerable attention due to the wide range of applications of these codes. Such constructions rely on non-special divisors of degrees $g$ and $g-1$. In this work, we investigate Kummer extensions defined by $y^m = f(x)$ with $f(x)\in\mathbb{F}_q(x)$ and establish an arithmetic characterization of non-special divisors whose support can contain non-totally ramified places. Based on this characterization, we explicitly construct non-special divisors of degree $g-1$ on the GK curve. Moreover, utilizing pure gaps, we explicitly provide several families of effective non-special divisors of degree $g$ on Kummer extensions with the same multiplicities. We then develop a general framework for constructing LCPs of algebraic geometry (AG) codes on Kummer extensions. By virtue of canonical divisors, we show that the security parameters of LCPs of AG codes can be determined within this framework, which also enables the construction of LCD AG codes. Finally, we illustrate our results with representative examples, including LCPs of codes on the GK curve and LCD codes on quotients of the Hermitian curve.