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Nothing if you can keep it running for more than 1 day in a row.

Not at all, they're actually pretty good cars.

Things to look out for:

Body rust. Check the fenders and wheelwells for excessive rust, especially if you live in a state that uses road salt.

Second gear. Take a test drive and give it a good solid drive, shifting through all the gears. If you hear second gear grind or make noise, then the syncro is worn. It's not a dealbreaker, you can live with it if you pause a second in neutral.

Electrics. Italian electrics are notorious. The major components generally work fine, but you'll have to keep up with the little minor things (like power windoes, interior bulbs, etc.)

I owned a 164 for a couple years, had all kinds of fun, and it was a very dependable car as long as I did the regular maintenance. Good luck!

I am surprised an Alfa of this age hasn't reduced to the std pile of red rust ... be very careful

Alfa of this vintage are fine cars when maintained and driven. The Bosch fuel injection is a fine reliable performer in the 2.0 l car.
When you look at this car for purchase look for the obvious signs that some one has had a fiddle with the wiring. when you sit down finally in the drivers seat flip every switch and work every electrical component to check for it's operation. It is true sadly when I bought my spider I had to chase loose grounds for several months before I got that all sorted. The Transmissions on most of these series Alfas had a weak second gear syncro so look for that "graunch" going through second. Rust on any car is an issue and if you live in an area where they salt the winter roads look very carefully at the sills and wheelarches. Also check the footwell floor boards for rusting.

There is nothing wrong with these cars and they are rewarding to drive and beautiful to see.

No, at least I don't think so, just keep the car in good shape, and you'll do fine, and keep it in good shape regularly...