Is my understanding of how this amplifier circuit works correct?
Sorry, but that's way off. 

Start with R1 and R2, and no input signal. Then R1/R2 form a voltage divider which sets the operating point for the transistor. It is assumed that the circuit is set up so that the base current is much less (&lt; 1/10) the unloaded current through R1/R2.

Now put a sine wave of (let's say) 3.9 to 4.1 volts on the input. The capacitor will indeed get rid of the 4 volt offset, but it will reference the AC to the operating point voltage. So let's say that +V is 6 volts, and R1 = R2. The operating point is 3 volts, and the capacitor will shift the input to 2.9 to 3.1 volts at the base of the transistor.
This voltage will be a sine wave, and this will cause the output of the transistor to be (more or less) a sine wave as well.