The short answer to your question is: Yes
Its generally advised to avoid making abrupt power changes to aircraft engines (piston and jet for that matter) to avoid surge and potential starvation issues. You are usually taught to apply and reduce power smoothly when learning to fly. Aircraft piston engines are not really all that different than older air cooled car engines (think VW and early Porsche (there was an aircraft certified Porsche motor actually)). As such they can (and sometimes do) suffer from the same pitfalls.
The difference you may see in aircraft engines is that they tend to be a bit better set up and less tinkered with than car engines.
Like any gas engine aircraft engines have been known to, on occasion suffer from vapor lock and fuel pump failure as well. The most dangerous and potentially serious issues in aircraft carburetors is ice build up due to high moisture and low temperatures at higher altitudes. Airplanes have carb heating systems that are manually controlled from the cockpit to solve this problem.
What would happen or what could be done if this scenario occurs?
Generally speaking this is a rare occurrence and often happens on the ground. Keep in mind that plane throttles are (during a routine flight) adjusted far less than a car throttle. In a car you may be going through a street with lots of stop lights causing you to go from idle to cruise/launch power frequently. In an airplane this really only happens once on take off and to an extend when reaching cruise. When you taxi out and line up you will advance the throttle to full/takeoff power to get in the air. You may then keep it there for your climb or elect for a slightly reduced climb power depending on the situation. You will climb to your cruising altitude and throttle back to between 55% - 75% power for cruise. Fly your flight and at some point you will get where you need to be and throttle back to your decent/landing power.
In the case of entering the runway and lining up if you starve the engine when you apply power you are assumed to be either not moving or barely rolling and should be safe. If you pull the power to quickly on decent and stall the engine you may be gliding distance from an airfield and or have time for a restart.
When it happened in cars sometimes you could not restart until the
residual fuel was burnt by cranking the engine for some time. Does
this apply also to aviation engines?
One neat thing that planes have which most cars done is mixture control. As a result you can pull the mixture control all the way out on a plane (how the plane is shut down), crank the engine, and draw the fuel out of the carb. Some cars will keep pumping fuel and filling up the carb while you crank (since you cant over ride the fuel pump). Generally a flooded car engine you have to wait for it all to evaporate out.
The question arises from the fact that the design of the engines are
quite old from my understanding.
Age has little to do with these issues. Fundamentally the internal combustion has changed little since its inception in terms of the way it physically works. While they have gotten far more efficient and reliable the physics are more or less what they have always been. Its our manufacturing technology that has made the engines better. Surging/starving an old engine and surging/starving a new engine will produce a similar issue. Aircrafts use older engines due to the complexity and cost of certifying anything for use in an aircraft. For what its worth Ford has been using the same basic V-8 for as long as some aircraft engines have been around likewise Porsche squeezed a remarkable 30 years out of the same basic air-cooled engine design in the 911 (911 up through 993 variant).
Side Note: As mentioned its expensive to get a new engine put into service. That being said there are common airplane engines out there (like the IO-550) that have fuel injection. As computers become more common place in aviation FADEC systems are becoming more common in airplanes. In the end of the day however, some of these choices come down to manufacture decisions. Some makers prefer to leave things in the hands of the pilots, in other words you can pull and advance the throttle how ever you like and the fuel/air delivery system will behave as such. Other makers have begun to put safe guards in place to prevent this, and they will (no matter how you operate the throttle) ensure proper advance and retard of all mechanisms as fast as possible should you operate the throttle vigorously.