How to find the charging time of a capacitor given a frequency and an amplitude
It won't "charge".
At 1 KHz, the voltage at the output will follow the input but with a small phase difference and small attenuation.
The capacitor plays little or no role at that frequency and with that resistor.

Beware that this changes totally if you would have e.g. a 2.84MHz. signal.



Sorry, with pulse I was assuming a sine wave which was switch on at T=0.
Added 1KHz, 50% duty cycle square picture. The capacitor charges but 'un-charges' as quickly.