How to measure capacitance without LC meter?
There is a solution using a microcontroller. Microchip wrote AN611 on it, but I'd recommend you this blogpost instead.  

How it's done: you start charging the capacitor and time how long it takes to get to \$\frac{1}{2}\cdot V_{in}\$. 



The 16F628A isn't necessary at all, you can use any microcontroller with a comparator.



With that time (the time it takes to get to \$\frac{1}{2}\cdot V_{in}\$), you can calculate the RC value of the RC circuit. You know R, it's 22K in the circuit above, it's constant. So you can calculate the value of C.