Since $(1,2)$ is a solution to both equations, plugging in $x=1$ and $y=2$ will give the values of $a$ and $b$.
$1 = \frac{1}{4} \cdot 2 + a$
$a = \frac{1}{2}$
$2 = \frac{1}{4} \cdot 1 + b$
$b = \frac{7}{4}$
So $a+b = \frac{1}{2} + \frac{7}{4} = \boxed{\textbf{(E) }\frac{9}{4}}$.