I just found a topic on Reddit where someone claims to have found a possible solution that worked for them. Here's what they had to say:
I just wanted to present this solution for anyone else that after the patch just got a black screen when trying to launch the game.
Possible fix #1 Head on over to "my documents" and rename the ME: Andromeda directory to something else. This forces Origin to re-download any save game files, while also creating new settings files. For some this has solved their problems.
Possible fix #2 (what helped me) Head on over to the installation dir of ME: Andromeda and remove the .exe files for Andromeda/Andromeda Trial. This forces Origin to re-download the .exe files so you have the latest version of them. For me, this was the solution- for some reason these .exe files did not update as intended after the patch.
Hope this helps anyone else facing the same problem! See you in Andromeda!"
Unfortunately, I have not had a chance to try this because my gaming rig (I'm on my toaster laptop) is disassembled right now, as I'm in the middle of upgrading my motherboard. Unfortunately, after redeeming the service plan on my last one, which was giving me troubles, and trading it in towards my new one, I realized that I left my CPU cooler bracket offset screws on the old board. Thankfully, I was able to call Microcenter and they were able to find my old board and remove the offset screws from it. So, they set those aside and I'll pick them up tomorrow.
Once I rebuild my gaming rig, I'll try the poster from Reddit's suggestions and will report back.
I hope EA takes us into consideration, because I did not pay for a broken game. I almost, almost wish I got the game on the PS4, because at least then it would have worked, despite being graphically inferior.
If I still can't get the game to work tomorrow, I will open a case with EA support directly.
~Agent Long Smith