W202 C180 Auto

Hi All. Last weekend I bought a 1995 C180 to use as a daily hack alongside my main car. However, it has got a few problems. The first is a non working water pump causing overheating. I am advised this is an easy fix, and so I am going to tackle this at the weekend. However, the car (an auto) cuts out when started. Is this common? And does anybody have any thoughts? Given that operating the sunroof or windows causes it to cut out, I wonder whether it is the battery? Or, as the water pump has had it, could it be that the alternator is not turning fast enough? My other thought was the fuel filter as, once the car is up and running, it doesn't cut out. Is this an easy fix? Any advice would be much appreciated - I am new to all this! thanks, Richard