Analog Integrator Sampling Rate
Sampling rate is only defined within discrete-time systems. So, there's really no sampling-rate equivalent that your integrator has.

What your integrator does have is finite bandwidth. It, for example, will not be sensitive for voltage variations at 2.4GHz.

The bandwidth off that circuit involves the characteristics induced by modeling the circuit based on ideal components, but also on limitations such as unintended (parasitic) inductance and capacitance of components and leads, and, most importantly: by the bandwidth of the opamp itself.

You can buy opamps with only a few Hz of bandwidth, you can buy them with multiple GHz of bandwidth, and very many values in between. Each of these have different strengths and disadvantages, so for a measurement system, choose wisely, based on an understanding of the requirements of your measurement system and signal, and at least a solid understanding of electronics and linear electrical networks; I'm not convinced designing a measurement integrator is trivial.

Generally, you might want to take a step back and actually ask over at electronics.SE how to measure a specific signal, describing exactly and in detail what you are observing, rather than assuming an integrator is the solution here. It might be - but it's equally likely you're not on the right track.