Can you provide a link to the heatsink you are referring too? The only one that comes up in a search is the big one for the desktop CPUs, in the LGA 1156, 775, 478, and AM2/3 sockets...surely you don't mean
this one do you?
If thats the case then no it will not fit the M350 case because it is 64mm tall while the M350 itself is only 62mm tall, but you have even less clearance than that once you have mounted the board in the case. Removing the fan would drop it down to 52mm presumably but thats not even the biggest problem..
The real problem would be in modifying the heatsink to fit the E350. These boards have embedded chips (the E350 and Hudson M1). You dont have a traditional mounting bracket like you do with desktop chips. All you have is 2 sets of tiny holes separated by about 40-45mm each. Plus with so many little capacitors and other components littered around the chips, you will have to cut into the heatsink and take notches out of it to fit around these components, which will be tricky with all those heatpipes. And thats before you even consider the difficulty you will have making a heatsink cover the M1 and the 350, or fitting a small heatsink just for the M1.
There are currently no after market heatsinks available to swap on these boards. You either have to live with what comes with the board, or modify something on your own. The fan that comes with this MSI board is pretty annoying, but you can remove it and install a better fan. The
40 mm Silenx for example is very quiet and provides enough airflow. Many of us have been able to remove the fan all together and just run passive with the stock heatsink. IF you have decent airflow this is a fine option. If you dont, then its worth it to add something like that silenx and just set the bios to reduce the fan speed until the chip reaches an unsafe temperate.