Wow, that looks super weird. Have never seen stuff like that before. Looks like a font issue at least, probably amongst others. Did the game function properly before you upgraded Wine? In that case I'd try to run Civ4 with the older well-functioning version of Wine. This can be done manually with Wine "bottles", or more automatically/easier with programs such as Lutris. You can try to install the basic Windows fonts into that "bottle", just in case it helps. And as usual, you need MSXML3 added for the game to function properly. No idea if lacking that can cause those issues, but it can (or will) cause problems with the map at times.
I have no experience with Arch sadly, so if it does anything special with Wine or whatever, you can try to look for solutions on more Arch-related sites I suppose.
Also, make sure the graphics card drivers are updated. You never know, it may impact stuff like this too.