While the engine will run just fine with any regular automobile type engine oil, I'd definitely choose something better suited for small air cooled engines.
Reason is, air cooled engines are exposed to a variety of conditions and temperatures that you don't find in modern liquid cooled auto engines. The high temperatures capable in air cooled engines can and will break down additives and shear the oil.
Choose the narrowest viscosity spread for the temperatures you plan on encountering. 5w-30 is okay in the winter, but I wouldn't use it in sustained high temperature operation. 10w-30 is okay for all season use, but a 30 weight will be more stable and maintain its viscosity better than a multi-grade in high ambient temps.
I use a 10w-30 HDEO, which has much a higher ZDDP and phosphorus count, which are important anti-wear additives, useful in this type of engine. Briggs and Stratton & Kohler branded oils are good choices as the level of additives present are far beyond that of an automobile oil, as there is no catalytic converter poisoning and efficiency to worry about.
I don't subscribe to the "thicker than manufacturers recommendation" ideas, but I bet Rotella 5w-40 HDEO or Mobil1 Turbo Diesel truck 5w-40 would be excellent choices as well if you're after a Synthetic.