I am a businessman who has a client in Weihai and Shanghai and I am arriving in Beijing

In will be applying for the VOA in Beijing.

Randal, unfortunately, you are not eligible for any kind of VOA and visa free transit schemes because you are traveling consecutively through several different Chinese transit zones. The only option for you is a rush visa service that has to be accomplished during the following two working days. If I were you, I would start to call the embassy in Dublin, right now. Trust me, there is no other option. I just hope that the embassy personnel will react promptly and help you in obtaining a visa.

Thank you can I ask if I was to get a 24 hour transit visa so Fly to Beijing then to Weihai and then to Seoul South Korea, Can I leave to airport in Weihai for say 3-4 hours before I depart to Seoul to meet my Chinese supplier.

Yes, that would be a solution. Now, you have to create an itinerary that will maximize your time in China, i.e. that it takes almost 24 hours from your first scheduled landing in Beijing, until your last scheduled departure from Weihai. One more important detail. You can use a single (through) ticket to Weihai, but you should definitely book a separate ticket from Weihai to Seoul in order to enjoy entry permit. In reality, this entry permit will be granted to you already in Beijing if you intend to fly through Beijing. You could also think about Shanghai as your entry point in China and then take a flight to Weihai.

So, your itinerary would be:

Europe - Beijing or Shanghai - Weihai - Seoul

Again, just keep in mind that the time you are allowed to spend in transit in China is calculated from your first scheduled arrival, until your last scheduled departure.