5V fan replacement
The MS-01 has a spare fan header, but it's a 5V header. The original n3rdware cooler comes with a 12V fan, so that's not suitable. Fortunately, a replacement is easy to source, an identical 5V fan was used in the Intel NUC 7. You can find it on eBay or Aliexpress if you search for Delta NS55B06.
Replacing the fan is easy: you unscrew 4 screws holding the shroud to the base. Carefully take off the shroud and unplug the fan. Unscrew the screws holding the fan to the shroud and switch the fans. Screw in everything exactly like it was before, and you're good to go.
Creating a fan adapter
The header you need to use in the MS-01 is a PicoBlade 1.25mm (Molex part number 510210400). Sometimes, the NS55B06 seems to come with exactly that header, but that is no certainty. If it doesn't, you will either need to make or find a suitable adapter. The pins are not an easy crimp, but you can find pre-crimped wires with housing on Aliexpress if you search for the Molex part number.
Hooking up the fan
The MS-01 comes with 2 spare fan headers near the M.2 drives. One has a convenient extension hooked up to it, the other one is right next to it but doesn't have the extension. If you don't need M.2 cooling, the first one is your best option.
If you do need M.2 cooling, you will have to use the second header. You will need to take off the front of the case (2 screws on either side), remove some tape from the USB ports and then remove the 3 visible screws mounting the motherboard. You can then pull up the motherboard just enough to plug in a cable in the header. Be very careful doing this as you don't want to damage the motherboard.
Carefuly route the cable to the top of the motherboard. The side of the case is probably the safest option, there is a little bit more room as opposed to the front.
Control the fan
Controlling the fan, based on the graphics card temperature, is the last thing to do. Regular software like MSI Afterburner will not be able to, but FanControl can control any fan based on any temperature sensor. You can read the documentation of the software for more information.