Is it possible to use a single ADC single S&H MCU for active power measurement?
What you are really saying is that there is a delay between measuring the voltage and current.

That delay represents a phase shift dependent on the frequency.  At low frequencies, that phase shift is acceptable.  At high enough frequency, it is not.

You have to decide what your maximum frequency of interest is, and how much error you can tolerate.  Compute the phase shift due to the delay at your highest frequency, then compute the error that causes.  If that exceeds the error you are willing to live with, then you need less delay between the two readings.

That all said, 1 kHz sample rate for computing real power and power factor for typical AC power is way too low.  You usually want to go 10x faster or more.  Remember that short spikes have significant high frequency harmonic content.  Again, you need to decide what your error limit is, but usually you want to do a decent job with the first 100 harmonics or so.

Added

You now say that your harmonic content is "low", and that the signals are "sinewave looking".  Note that this is NOT the case for the current waveform you show.  Those spikes add significant harmonic content.

If at least one of your signals has low harmonic content, then you can measure it first and adjust it in time to the second signal.  Usually the voltage is better behaved and more sine-like.

You could, for example, do a polynomial fit on the last few samples, then use that to extrapolate the signal to the same time the other signal is actually measured.  Or, if you can measure the two signals alternately, wait one sample so that you can use the average of the last two samples of the low-bandwidth signal to multiply by the other signal.  That's basically interpolating the two adjacent samples to approximate the value at the same time as the sample of the high-bandwidth signal.

In short, there are various ways to filter and interpolate multiple samples of one signal to estimate its value at the sample time of the other signal.