It seems that updating with anything besides Windows is a no-go because of the structure of the programs. Which is unfortunate. Because of this it seems this method of updating on Linux does not work. I'll try to find a way in my spare time, but don't get your hopes up. For now re-installing the prefix and/or MTGA is your best bet.
Since the original MTGALauncher.exe doesn't work on Linux without significant botching, I translated it to python. Instead of windows calls it'll make calls to your WINEPREFIX where MTGA is installed.
This is pretty much untested, this might fuck your MTGA-install (WINEPREFIX maybe? But might aswell delete the prefix if MTGA-install is fucked, no?), so make sure you have a back-up (Just duplicate your prefix folder). I doubt we'll be needing this, but nonetheless: I'd love to hear about it, even if it went successfully.
Make sure your MTGA.exe tells you your game is out of date and you /NEED/ to run MTGLauncher.exe, otherwise this'll just be a waste of time (You see this /AFTER/ logging in and trying to actually get into the game.).
Once again! Make sure your MTGA.exe tells you your game is out of date and you /NEED/ to run MTGLauncher.exe, otherwise this'll just be a waste of time (You see this /AFTER/ logging in and trying to actually get into the game.).
Place the MTGALauncher.py next to the MTGA.exe and execute it in a terminal (./MTGALauncher.py)
Wait for it to finish (it'll launch MTGA.exe for you) and see if MTGA.exe will now let you log in. Please do report any issues you have as it'll undoubtedly help the cause.