Measuring the tRNA maturation activity of RNase P by a high-throughput-compatible method has been reported, but I presume this was done using purified enzyme:

"Fluorescence-Based Real-Time Activity Assays to Identify RNase P Inhibitors"

Chapter Fluorescence-Based Real-Time Activity Assays to Identify RNa...

It might not work in a crude cell extract because of the presence of other RNase enzymes that might catalyze the same reaction. Therefore, an immunological approach (e.g. Western blot) might be more useful. For that you will need an antiserum or antibody specific for the RNase P of interest. You will probably have to make the antiserum yourself or get some from another researcher, because it is unlikely to be commercially available for bacterial RNase P.

Another approach to find out whether a complex mixture contains RNase P (or any other protein) is proteomics. This would require working with a specialist in this technique.

A third approach would be to fractionate the extract by chromatography, test each fraction for RNase P-like activity using an assay such as the one described in the paper above, and further purify the active fractions by additional chromatographic methods until you have a sufficiently pure fraction to analyze by mass spectrometry for the presence of RNase P. You might be able to follow published purification steps.

https://academic.oup.com/nar/article/29/18/3848/1058129

Thanks for sharing these very useful information. Thank you so much.

Is there any that can confirm its binding with arsenite or arsenate?

You could probably use isothermal titration calorimetry for that purpose, if you have purified RNase P in substantial quantity. If arsenate or arsenite inhibits the catalytic activity, then you could demonstrate binding by measuring inhibition.