Traveling from Munich to Amalfi

I'd take the Circumvesuviana train from Naples to Sorrento and catch a bus to Amalfi from there. http://www.theamalficoast.net/amalfi/english/bus_ferries.html

You hadn't mentioned what date you were going to do this route. You are right that a flight will be better. If you can get a direct flight into Naples that would definitely be faster and probably cheaper. Otherwise a good alternative would be Rome. You would then train down to Sorrento and then taxi or boat to Amalfi.

Look on www.skyscanner.com for budget flights. I looked on an arbitrary day in the end of June and I saw that Meridiana had an early direct flight from Munich to Naples for about $129 per person.

If you fly into Rome instead, you can take a high speed train down to Naples taking about 1.5 hours. Then instead of taking a taxi to Amalfi (which would be expensive), you can take the Circumvesuviana train or a boat to Sorrento. Once in Sorrento you could do a boat, bus, or taxi to Amalfi.

If you haven't booked your hotel there yet, you might consider staying in Sorrento. It is a pretty town and the transportation there is much easier than it is to Amalfi. I hope this helps some.

Bev, many people advise staying in Sorrento and daytripping to the Amalfi Coast. However, Sorrento is not part of the Amalfi Coast. In my opinion there is no substitute for Positano or Ravello in that area.

Zoe is 200% right. If you want to go to the Amalfi Coast, you should stay in the Amalfi Coast. Some people prefer Sorrento, which is not on the Amalfi coast, because it's logistically more convenient for visiting Pompeii and maybe Naples, but to me Positano is the place.
Yes, flying from MUC to NAP is best both time-wise and probably money-wise. To save money you could also take a CurreriViaggi bus from the airport directly to Sorrento (for €10). Then from Sorrento you can take a SITA bus to the Amalfi coast. A private car transfer from NAP to the Amalfi coast is probably close to 100 euro.