Can a Skylane fly without electricity?
The engine wouldn't fail. Avgas powered engines use 2 magneto systems which are independent of the battery and alternator, so if your alternator dies or you lose the electrical system it will run as if nothing happened. Starting it does require the battery and electrical system though to run the starter motor. Aviation engines with computerized ignition systems have backup batteries so they will also not fail due to an electrical system malfunction. 

As for the "6 pack" cluster of instruments 3 are not powered at all: Airspeed Indicator, Vertical Speed Indicator, and Altimeter. 3 are powered, although there is no hard and fast rule the Attitude Indicator (AI) and the Heading Indicator (HI) are usually vacuum powered from the engine and the Turn Coordinator is usually electrically powered. There are many exceptions to this. 

So keeping the electrical system off doesn't affect the primary instruments much, provided the airplane stays in VFR. What captain cheapskate won't have for sure is any radios, transponder, GPS, ADF, VOR, or any other communication or navigation aids. 

All these things you could probably do without on a completely VFR flight, provided you don't need communications. However, the engine temperature and oil pressure gauges, EGT gauge, and fuel gauges all work using electricity as well, and I'd really hate to lose those. A typical steam gauge Cessna isn't going to have an electrical fuel pump, but many piston singles do. Running without the electrical system means you'd lose that to, and that's considered an important safety system to back up the mechanical one in case of failure.