The charging time of capacitor
(considering this might be homework)

Try using some Norton->Thévenin transform on the current supply and one of the resistors, it will make your life way easier and the solution will be pretty much that. 

But, considering you find that counter intuitive. Think of what happens when the capacitor completely charges. If the capacitor is charged that means no current goes through the capacitor branch, so all the current \$ I \$  goes through \$ R_1\$, from that you can figure out to what voltage the capacitor will charge (\$ V_C = I R_1\$). Now you can use KCL and KVL to have,
$$ I = i_{R1}(t) + i_{R2}(t), $$
$$ R_1i_{R1}(t) = R_2i_{R2}(t) + V_{C}(t), $$
$$ V_{C}(t) = \dfrac{1}{C}\int^t_0{i_{R2}(t)dt}. $$
That should be enough to solve it.