TBH I hate desktop linux as it always has issues, but I swear by using linux for servers so I've got a little experience with it and I know there are a few others in the osu! community
osu! on Linux seems to be a hit and miss at the moment. First of all, it depends on wine, since there currently is no released osu! client officially for Linux (
yet). That said, what might be a down point for me might be an up point for you, as getting it running is kindof part of the point to running desktop linux, which is hacking around to learn. There's plenty of resources around to get it working, and if you're running Ubuntu I had it work pretty much out of the box on Ubuntu 16.10 (16.04 should work no differently, just install fonts, .net 4 and run it which is all I did).
Getting it to run portably should, theoretically, be no different than running it from your hard drive, (only difference is it's on a thumb drive) just make sure you do it in a 32bit prefix as I doubt it'll work in a 64bit one and odds are, a fresh install of wine the default prefix you make will likely be 64bit. From there it's just a matter of launching your portable osu! in the prefix, which you can either do from terminal or make a simple .sh script to do it.
Why not try it out and see? If you run into any trouble I'm sure there are many in the community willing to help (just check the osu! on linux threads) or post back here/shoot me a message, if I see it I'll try to pitch in best I can.