Hmm, so
https://osu.ppy.sh/p/download mentions you now need .NET Framework 3.5. Doing winetricks dotnet35 however installs 2.0, 2.0SP1, 3.0, and 3.0SP1 .NET Frameworks. And even after doing this, the installer still complains of the need of .NET 3.5. In other words, the new installer doesn't work at all it seems (using wine1.6 from Ubuntu's repositories).
My recommendations are to either use the old installer (perhaps someone could mirror it?), or just grab osume.exe and run that to grab the files. If you use osume,
grab it from here, make the osu! folder in
Program Files, run osume, let it update, then proceed as normal (my full instructions a page or two back still work; still required to use 32-bit wine prefix)
Both osume.exe and osu!.exe run just fine with just dotnet20 still, so I wonder about the 3.5 reliance. Can still use gdiplus if you want the fonts to look nicer, but note you need to download about 1GB or so now when using winetricks (wtf?).
Same old Combo Fire and scoreboard name glitch still exists it seems on my hardware (Lubuntu 13.04 + fglrx (13.8 beta)).