Well archaeology happens to be my field as well and have been using GIS for about 8 years now. Are you specifically looking for ArcGIS tutorials or GIS in general? I can definitely recommend this book. If you need help with anything specific, you can throw me a dm or stack overflow is your best friend.
Does it need to be done through Arc? Mapbox has sheetmapper, which updates automatically from a Google sheet.
If I were doing this, I'd write some function that pulls data from your data portal (is it available through download? API?) in bash/python, transforms the data to whatever variables/name conventions you need using SQL, extract the area of interest using PostGIS ST_WITHIN or DWITHIN and then populates the sheet. I'd probably rely on a python library to populate the Google sheet.
I'm not sure how frequently the data is actually added by your city-- is it daily? monthly? quarterly? Because if you want things to run daily, it makes sense to use some sort of automated trigger that will run independent of your input (AWS lambda is the first thing that comes to mind), but if it's monthly, its possible you can store your script locally and maybe even have it as a automated scheduled task.
I don't know if you are still having this problem, but I ran into a similar issue at work with trying to merge DEM raster images. I found the ArcGIS plugin was not working all that well and it was giving errors/odd color combinations.
Anyway, I was able to successfully merge the raster DEM images together using qgis. Hopefully this helps you and you are able to produce a nice clean product. It worked well for me!
http://www.qgis.org/en/site/forusers/download.html#
http://www.qgistutorials.com/en/docs/raster_mosaicing_and_clipping.htm
There is a newer version of this text book the op might want to know about here: https://www.amazon.com/dp/1589484991/ref=cm_sw_r_cp_awdb_btf_t1_j6SHFbHEE4P2E
That one is great but I think the op might even want the more advanced text book that goes with it here:
https://www.amazon.com/dp/1589486188/ref=cm_sw_r_cp_awdb_btf_t1_B-SHFbE0J3Q1X
I can't recommend either book enough.
I suggest checking out
It was a very useful starting point for me. It will cover all of the basics, plus how to set up your IDE.
I use Pycharm for all my development, but you can use whatever IDE you feel comfortable with.
All the ESRI sites have examples of scripting at the bottom of each tool's page. A quick Google search of a tool plus Arcpy should give you what you are looking for.