I have the price per square meter of around 13000 properties ; for the other 8000 of my shapefile the price is missing. The easiest option is to transfer the average price per square meter to the missing properties. However, I believe it would be more realistic if a sort of neighbour effect would be considered in the calculation. I thought therefore to use interpolation tool. I am not very experienced in it, I have read the tutorials and explanations, but it is still to me not very clear what tool would be more appropriate for my case, and with what settings. Would you be able to advice on this?
I'm not sure which interpolation method would work best in your case, but I do have a couple of suggestions as to how you can figure it out.
First, do a bit of research on how the various interpolation methods work to identify which you think would best suit your data. For example, the IDW method is a spatial averaging technique that outputs data that will be bounded by the input data (e.g. the highest value in your output will be no higher than the highest value in your inputs). If that assumption makes sense for your application then IDW might be a good approach.
Second, you can use your existing data to perform some analysis on which method may work best. If you randomly extract a subset of your known data set, the table with 13,000 records, you can use them to benchmark how well different approaches work. For example, randomly remove 3,000 records from your properties data, use the remaining 10,000 records to interpolate pricing, then compare the actual values in the 3,000 records to the interpolated model. This will allow you to quantitatively compare different approaches and identify which one works best for your situation.
Kriging is easily the best function for this, especially for hedonic features such as home price, sq ft. etc. The geostatisical wizard in ArcGIS will be able to do this for you. Make sure you read the background on kriging as there are a number of parameters that should be input to your model.