Most parametric and nonparametric tests assume sample independence...so if your sample variable is spatially autocorrelated then ANOVA could be problematic. If you provide a bit more detail, I might be able to point you in the right direction.

If your data are normally distributed so that if you can do an analysis of variance for comparison of means and further analysis with a geostatistical kriging interpolation. If your data does not have a normal distribution, you can do an analysis of variance to compare medians, also you can make a indicator kriging interpolation if you have a threshold (cut off).

A compariosn of means, either parametric as ANOVA or non-paramatric as for example Kruskal-Wallis on data points grouped by some spatial container like administrative boundaries should be sensible. The choice must be informed by your sample sizes for each class, the distribution etc. as has been mentioned. But if you first interpolate by Kriging and then use that as input to the ANOVA etc. that would be a problem as far as I understand.

You can apply ANOVA if your data fulfill the respective requirements. You need approximately Gaussian-distributed residuals (because ANOVA is based on testing Fisher-distributed values), homoscedasticity within the groups (otherwise the groups are inhomogeneous and you should revise them) and the data should be iid (which means not autocorrelated and equally distributed over your sample area). As Andreas already mentioned, you should not use a full sample after applying Kriging interpolation. This would create an inherent spatial autocorrelation, infringing the requirements. So, if you want to apply ANOVA use some test for normality, assure that your data points are identically distributed and classify them into reasonable classes.