I don't know anything about Mame but Ubuntu is a pretty simple install these days. the primary install is just a GUI like any other. However, there are some extras you probably need afterwards but the 'Interwebs' make that so simple these days.. Google a couple of "Things to do after installing Ubuntu" pages for very straight forward instructions and pick out the items that seem relevant.
There may be some proprietary drivers for Mame and/or the joystick console that I know nothing about but I'm betting Google has that too from people who have already done it. Before you install Linux, check out which Distro (Ubuntu, Debian, Mint, etc.) that seems to have the most support and people behind what you are doing. Ubuntu seems to have the most, easy to find support, in general, due to it's huge user base but maybe there are more popular versions for your specific purpose.
My Linux skills are mediocre are best but it is what I run on my home PC now. So if I can help with the simple stuff, I'll be glad to try and I know there are a couple of real experts on this board who helped me. So you should be covered.
