Ok, I tried patching the port itself by fiddling with its insides and aparrently apps don't like it when you do that. No surprise when you consider that this is all in all still a mac program that adores to modern OS X rules instead of ancient "anything goes" DOS rules.
My advice is to play the game in Boxer instead of using GoG's port. First of all download Boxer:
http://boxerapp.com Now right-click the app -> Show Package Contents -> Resources and move the Ultima 3.boxer file somewhere you want (Boxer creates a DOS games folder for you, that's a good place). Now download the patch and unzip it. Inside the folder you should find lots of files (you might have to go one or two folders deep to get there). We will need these files for the patch.
Now right-click Ultima 3.boxer -> Show Package Contents ->C Ultima3.harddisk and copy all of the files mentioned above into that folder. Launch Ultima 3 and you should get a DOS promt. The next steps are exacly as described in the readme of the patch:
1) copy-paste BINPAT.EXE U3UP-30.PAT into the promt, hit enter and wait for the patching to finish
2) type U3CFG.EXE to change settings (the defaults are best in my opinion)
3) type SETM.EXE to choose your music card, use Sound blaster when in doubt. Boxer supports Roland, so you can use that if you find the ROMs on the internet
You can play Ultima 3 by typing ULTIMA3.COM into the prompt. If Boxer asks whether you want to launch that program automatically say yes and from then on all you need to do is bouble-click the box and you can play.
EDIT: If for whatever reason you
really insist on using the port instead of Boxer, maybe Alun Bestor, the author of Boxer who also worked with GoG on creating the tool for these standalone ports, can help you out.