Will a low-pass filter make zero-crossing detection less precise?
Zero-crossing is the wrong way to do this, as it only uses two samples on the period (ignoring the rest) and is thus very sensitive to noise.

If your signal is a sine of constant amplitude (at least over a few periods) then the best way is to acquire the signal, and multiply it in the digital domain with sine and cosine waveforms. This IQ demodulatoin gives you a complex number, whose angle allows measuring the phase shift with extreme precision. 

For example, it is possible to measure nanosecond phase shifts between two sine waves using a simple PC soundcard, as long as SNR is adequate.

This is equivalent to computing only one bin on a FFT.