capacitance measurement circuit
As you've found out, without R1, there's no DC path to the amplifier inverting input. Input leakage current will take it off to rail (+ve or -ve depending on the sign of the current) and the amplifier will stop working.

There are two principal options for the value of R1

a) R1.Cf >> input period. Then the gain is dominated by the value of Cf, and you don't need to know the exact frequency. R1 must be small enough to source all of the input leakage current, but that's generally fairly easy for most opamps with a 1M or 10M resistor.

b) R1.Cf is not >> input period. Then you need to allow for R1 and know the frequency if you are calculating the gain. If you are simply making a capacitance meter, and you calibrate the scale with a known capacitor, and keep the frequency constant, then you can ignore their actual values.