How long does Marsala wine last, after opening?
Marsala is a fortified wine, that is, a wine to which extra alcohol has been added. 

Therefore, you can store Marsala outside of the fridge, in a cool place (15-20 °C), away from direct light. In these conditions a bottle can sit there for several weeks/months with no obvious degradation, as you would expect for fortified wines. 
Generally it is suggested to drink Marsala or similar wines, such as Porto or Banyuls, in the following 4/5 months, but I had bottles opened for a year or so and they were still good, although the flavour faded a bit.

As for Marsala all'uovo, it has pasteurized egg yolks in it, and even home-made recipes call for a pasteurization step.
In general I would store home-made one in the fridge to be on the safe side, but do not serve it too cold. Commercial ones may also have preservatives (aside from the added alcohol of Marsala) in them, and are safe to leave in a cool place for several weeks.