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.

Another answer, given that you have a frequency counter, would be to build an oscillator using the 555 timer, and measure the frequency. Switch in a few standard resistors (1k, 10k, 100k etc) and check with a known capacitor how close the "frequency calculation" formula is to reality.