How do I work out the time when given current in a RC circuit?
This is a rather poorly set question- no information is given as to the initial condition of the capacitor. If the 10uF cap happens to be charged to 12V the current will be 0 before and after the switch closes.  

Anyway, what they want you to do is to assume the initial voltage on the capacitor is 0V. 

You should then be able to write down the equation for voltage on the capacitor as a function of time (memorize this- it's the solution to the differential equation if you want to do it from first principles). 
$$
V_c(t) = 12 (1 - e^{-t/\tau})
$$
where \$\tau = RC\$. 

Since you know that the current is \$(12V - V_c(t)) \cdot 2.2k\$ you can solve for when current is 2mA. 

Alternately you can recognize that 
$$i(t) = I_0 e^{-t/\tau}$$ 
where \$I_0 = 12V/2.2K\$ (you have this) 

so
$$
t = -\text{ln}(2mA/I_0)* RC
$$
and the answer is 'b' 22ms

To get the above line, divide both sides by \$I_0\$, then take the ln() of both sides and solve for t.