I run Portable Firefox from Portable Apps on a fairly regular basis but when I find myself on a machine where I haven't got Admin privileges I need to run another program (in this instance 7-zip), again from Portable Apps.
I download http://fpdownload.macromedia.com/get/flashplayer/xpi/current/flashplayer-win.xpi, open the flashplayer-win.xpi with 7-zip and extract the flashplayer.xpt and NPSWF32.dll files to the \FirefoxPortable\Data\plugins folder. Start up Firefox again and all should be well and you can browse YouTube to your hearts content.
I've learnt this trick from a number of sources and used to use it to get the now defunct Adobe SVG plugin to work nicely with Firefox... I did however refresh my memory using the answer posted on Acid-Labs: Installing Flash in Portable Firefox with no installer... particularly the one by Vishwesh Sharma (No. 12).
This got me thinking about installing the Adobe plugin in Firefox 3.5 and, despite following the instructions from 2006, I couldn't get it to work... such a shame Adobe stopped updating it as it's scripting support was excellent.