Low power rectification of AC component of modulated current signal
Are there any obvious problems with the circuit, that may lead to the
  observed drifting behavior?



The minimum supply for an LPV821 is 1.7 volts and you are expecting it to work above 1.6 volts.
Your AC current is 1 uA and this produces a voltage (across the 20 ohm resistor) of maybe 28 uVp-p. Given that your op-amp is running without feedback AND it has an input offset voltage of about 10 uV, you will certainly get poor performance.
Given that you are creating a voltage signal at the op-amp's +INPUT that rises above and falls below 0 volts by about 14 uV, the negative excursions exceed the common-mode input range for the op-amp as specified in the data sheet.
Guaranteed data sheet specifications are when operating from a supply voltage of 1.8 volts. The front page says 1.7 volts but there's no guarantees at this level.
The input voltage noise density is 215 nV/\$\sqrt{Hz}\$ and, without any filtering on the op-amp output this is likely to smudge your signal. Assuming a bandwidth of 5 kHz (GBWP = 8 kHz), the equivalent input noise is 15.2 uV RMS and and is a lot bigger than the signal you are trying to amplify.
Running open-loop (no feedback) means that any signal at the output will be shifted by about 90 degrees. See figure 11 in the data sheet. Given that you want the phase to be the same as the input signal this appears to be a showstopper.